Proof. We give the decomposition argument to specify precisely which parts of the Cartan action may be nonsemisimple; see also (Cornulier and Tessera 2017, sec. 4.D) for the Cartan grading. For \(v,u\in\mathfrak d\), nilpotence of \(\mathfrak d\) gives \[(\mathop{\mathrm{ad}}_{\mathop{\mathrm{End}}(\mathfrak g)}(\mathop{\mathrm{ad}}v))^b(\mathop{\mathrm{ad}}u) =\mathop{\mathrm{ad}}((\mathop{\mathrm{ad}}v)^b u)=0\] for sufficiently large \(b\). Consequently \(\mathop{\mathrm{ad}}u\) preserves every generalized eigenspace of \(\mathop{\mathrm{ad}}v\). Indeed, between two primary blocks with distinct eigenvalues, the commutator with \(\mathop{\mathrm{ad}}v\) is invertible, so an operator killed by a power of that commutator has no such block. All eigenvalues are real by real triangulability. Successively taking the primary decompositions for a basis of \(\mathfrak d\) therefore gives simultaneous invariant primary spaces. On each such space all diagonal entries of a triangular representation agree on that basis, and hence agree as linear forms on \(\mathfrak d\). Their common form \(\gamma\) is a character, since diagonals of commutators vanish. This proves the direct sum decomposition into the spaces in (2).

The derivation identity shows that weights add under brackets: applying \(\mathop{\mathrm{ad}}v-(\gamma(v)+\eta(v))\mathop{\mathrm{id}}\) repeatedly to \([x,y]\), with \(x\in\mathfrak g_\gamma\) and \(y\in\mathfrak g_\eta\), gives a binomial sum involving powers of \(\mathop{\mathrm{ad}}v-\gamma(v)\mathop{\mathrm{id}}\) on \(x\) and of \(\mathop{\mathrm{ad}}v-\eta(v)\mathop{\mathrm{id}}\) on \(y\). Every summand vanishes at a sufficiently large power. Nilpotence of \(\mathfrak d\) gives \(\mathfrak d\subseteq\mathfrak g_0\). If this inclusion were strict, the induced action of \(\mathfrak d\) on \(\mathfrak g_0/\mathfrak d\) would consist of nilpotent operators. Engel’s theorem would give a nonzero fixed class. Any representative of that class normalizes \(\mathfrak d\), contrary to self-normalization. Thus \(\mathfrak g_0=\mathfrak d\).

The nonzero weight spaces are \(\mathfrak d\)-invariant. Their generated subalgebra \(\mathfrak n\) is therefore \(\mathfrak d\)-invariant, and \(\mathfrak g=\mathfrak n+\mathfrak d\) makes it an ideal. If \(\gamma\ne0\), choose \(v\in\mathfrak d\) with \(\gamma(v)\ne0\). The restriction of \(\mathop{\mathrm{ad}}v\) to \(\mathfrak g_\gamma\) is invertible, so \(\mathfrak g_\gamma\subseteq[\mathfrak g,\mathfrak g]\). In a simultaneous real triangularization of \(\mathop{\mathrm{ad}}(\mathfrak g)\), every element of \(\mathop{\mathrm{ad}}([\mathfrak g,\mathfrak g])\) is strictly upper triangular. Engel’s theorem applied to the derived algebra proves its nilpotence, and hence that of \(\mathfrak n\).

The images of the generating nonzero weight spaces span \(\mathfrak n/[\mathfrak n,\mathfrak n]\), and its induced weight decomposition has no zero summand. Choose a homogeneous basis of this quotient and homogeneous lifts \(X_i\). These lifts generate \(\mathfrak n\): if \(\mathfrak h\) is their generated subalgebra and \(\mathfrak n^j\) denotes the lower central series, then \(\mathfrak n=\mathfrak h+\mathfrak n^2\). Induction, using brackets, gives \(\mathfrak n=\mathfrak h+\mathfrak n^j\) for every \(j\), and nilpotence concludes the argument.

We now pass from Lie algebras to groups. Dixmier’s real-root criterion implies that \(\exp_G:\mathfrak g\to G\) is an analytic diffeomorphism (Dixmier 1957, Theorem 3). The inclusion of either nilpotent subalgebra integrates from its simply connected Baker–Campbell–Hausdorff group, and this homomorphism is the restriction of \(\exp_G\) in exponential coordinates. It is injective, with closed image, because \(\exp_G\) is a diffeomorphism and each subalgebra is a closed vector subspace. Injectivity of \(\exp_G\) also gives the stated description of \(N\cap D\). This proves the claims about \(N,D\), including normality of \(N\).

The group law in the domain of \(q\) is \[(n,d)(n',d')=(n(dn'd^{-1}),dd'),\] which verifies the homomorphism assertion. Its derivative at the identity is \((x,v)\mapsto x+v\), which is onto. Its image is therefore an open subgroup of the connected group \(G\), so it is all of \(G\); translation also makes \(q\) a submersion everywhere. The kernel formula follows directly from \(nd=1\). Local smooth sections of a submersion, restricted to relatively compact neighborhoods, give compact sets of lifts; a finite cover proves the claim for an arbitrary compact set.

If \(\mathfrak n=0\), then \(\mathfrak g=\mathfrak d\) is nilpotent. Conversely, in a nilpotent \(\mathfrak g\) all adjoint operators are nilpotent. The decomposition just proved then has only weight zero, so \(\mathfrak g=\mathfrak d\) and \(\mathfrak n=0\). ◻

Proof. We first bound the coordinates of a short path. For the reverse estimate, we construct uniformly bounded products of degree-one exponentials in the rescaled groups. The remaining assertions follow from these coordinates and the Haar Jacobian.

A path of length at most \(t\) splits its endpoint into a product of at most \(C(1+t)\) increments from a fixed compact neighborhood. Their logarithms lie in a fixed bounded set. In a product of \(a\) exponentials, the degree-\(b\) part of its formal logarithm has a sum of absolute coefficients at most \(C_b(1+a)^b\). To see the bound, expand the product of formal exponentials through degree \(s\) in the noncommuting variables, and then use the finite expansion of \(\log(1+u)\) in those degrees. The degree-\(b\) coefficient sum of the product is at most \(a^b/b!\); the finitely many compositions of \(b\) appearing in the logarithm give the asserted bound. Applying the degree-\(b\) Dynkin projection, which sends an associative monomial to its iterated bracket and acts by multiplication by \(b\) on homogeneous Lie polynomials, converts this expansion to brackets with only a degree-dependent coefficient bound. A bracket involving more than \(j\) increments belongs to \(\mathfrak d^{j+1}\) and has zero \(V_j\) projection. Evaluating the remaining finitely many types of brackets on the bounded logarithms therefore bounds this projection by \(C_j(1+a)^j\). This proves (7).

For the converse, we first record the limit of the transported laws. The inclusion \([V_i,V_j]\subseteq\bigoplus_{k\ge i+j}V_k\) shows that the coefficient of a bracket from \(V_i\times V_j\) into \(V_k\), when transported by \(\delta_r\), is multiplied by \(r^{i+j-k}\). For \(r\ge1\) these coefficients are bounded and converge to the associated graded bracket. The finite Baker–Campbell–Hausdorff formula proves convergence of the group laws and all their derivatives on compact sets. Denote the transported group by \(D_r\); its underlying exponential coordinate space is the fixed vector space \(\bigoplus_jV_j\).

We now produce uniformly bounded products of degree-one exponentials covering a fixed coordinate neighborhood. Choose a basis \(e_1,\ldots,e_a\) of \(V_1\). These vectors generate the graded Lie algebra: by the definition of the lower central series, the quotient \(\mathfrak d^j/\mathfrak d^{j+1}\) is spanned by brackets of length \(j\) of classes in \(\mathfrak d/\mathfrak d^2\). In \(D_\infty\), let \(H\) consist of all finite products of \(\exp_\infty(t e_i)\) and put \[L=\mathop{\mathrm{span}}\{\mathop{\mathrm{Ad}}_h e_i:h\in H,\ 1\le i\le a\}.\] This space contains the \(e_i\) and is invariant under every \(\mathop{\mathrm{Ad}}_{\exp_\infty(t e_i)}\). Differentiation makes it invariant under \(\mathop{\mathrm{ad}}e_i\), so it contains all their brackets and equals the entire graded Lie algebra. Choose \(b=\dim D\) such conjugated vectors \(\mathop{\mathrm{Ad}}_{h_\ell}e_{i_\ell}\) forming a basis, where each \(h_\ell\) is a fixed finite product of degree-one exponentials. The map \[\Phi_\infty(t_1,\ldots,t_b) =\prod_{\ell=1}^b h_\ell\exp_\infty(t_\ell e_{i_\ell})h_\ell^{-1}\] is the identity at \(t=0\) and has these basis vectors as its derivative columns there.

Use the same fixed words for \(h_\ell\) in \(D_r\) to define \(\Phi_r\). The maps \(\Phi_r\) converge with their derivatives to \(\Phi_\infty\) on compact parameter sets. The inverse function theorem therefore supplies a fixed parameter ball and a fixed coordinate neighborhood \(U\) of the identity contained in their images for all sufficiently large \(r\). More explicitly, choose a parameter ball on which the derivative of \(\Phi_\infty\) differs by less than one quarter of the smallest singular value of \(D\Phi_\infty(0)\) from that derivative; the corresponding estimate with one half holds for \(\Phi_r\). The usual contraction proof of the inverse function theorem then gives one common image ball. Every element of \(U\) is thus a product of a fixed number of \(\exp_r(c e_i)\), with all \(|c|\) bounded by one constant. The conjugators here are themselves fixed words in these exponentials.

A fixed number of powers of \(U\) covers the coordinate box \(\{\sum_jv_j:\|v_j\|\le1\}\), uniformly in \(r\): choose an integer \(M\) large enough that every vector \(v\) in this box satisfies \(v/M\in U\), and use \(\exp_r(v)=(\exp_r(v/M))^M\). Under \(\Delta_r\), each degree-one factor becomes \(\exp_D(rc e_i)\), whose intrinsic length is at most \(r|c|\|e_i\|_D\), where the last norm is the Riemannian norm at the identity. The resulting products have length at most \(Cr\). Enlarging \(C\) handles the remaining compact interval of \(r\ge1\). Given arbitrary \(v_j\), apply this bound with \(r=\max(1,\max_j\|v_j\|^{1/j})\) to obtain (8).

For the equicontinuity assertion, the logarithm of the difference of \(z_r\) and \(z'_r\) in the transported law is a vector \(\xi_r=\sum_j\xi_{r,j}\) tending to zero. The logarithm of the corresponding difference in \(D\) is \(\delta_r\xi_r\). Thus (8) gives \[\frac{\mathop{\mathrm{dist}}_D(\Delta_r z_r,\Delta_r z'_r)}r \le \frac Cr+C\max_j\|\xi_{r,j}\|^{1/j}\longrightarrow0.\] The compact length bound follows from the same coordinate estimate. The converse compactness assertion follows from (7) after dividing the \(V_j\) coordinate by \(r^j\). Equation (10) follows directly from convergence of the transported product, since \(\delta_r(tv)=rtv\) on \(V_1\).

Finally, the differential of the exponential map, after left translation to the identity, is a polynomial in \(\mathop{\mathrm{ad}}v\) with constant term \(\mathop{\mathrm{id}}\). Since \(\mathop{\mathrm{ad}}v\) is nilpotent, this polynomial has determinant one. This proves the claims about Haar measure on both nilpotent groups. The linear coordinate map \(\delta_r\) has determinant \(r^Q\). The coordinate bounds (7) place a radius-\(t\) ball in a product of Euclidean balls with radii \(C(1+t)^j\), proving the upper volume bound. Conversely, (8) puts the coordinate box \(\{\sum_jv_j:\|v_j\|\le(t/(2C))^j\}\) inside the radius-\(t\) ball when \(t\ge2C\). This box has volume a fixed positive constant times \(t^Q\). Decreasing that constant handles \(1\le t\le2C\), and proves (11). These two bounds give doubling for \(t\ge1\). At sufficiently small radii a smooth coordinate chart and the Riemannian metric give volume comparable to \(t^{\dim D}\); the remaining compact interval of radii is handled by monotonicity and positivity of ball volume. This proves (12). The proof used only simple connectivity and nilpotence, so it applies without change to the group \(L\) in the last assertion. ◻

Proof. The restrictions of the diagonal characters \(\chi_j\) to \(\mathfrak d\) are exactly the generalized weights of its action on \(\mathfrak g\), with their multiplicities. All \(\chi_j\) vanish on \(\mathfrak n\) by Lemma 5. In particular every Cartan weight vanishes on \(\mathfrak d\cap\mathfrak n\), and, being a character, also on \([\mathfrak d,\mathfrak d]\).

Homogeneous bracket words in the \(X_i\) span \(\mathfrak n\). Their weights are sums of the \(\alpha_i\), whereas every \(\alpha_i\) itself occurs. The weights on \(\mathfrak d=\mathfrak g_0\) are zero. It follows that \[\mathfrak k\cap\mathfrak d=\bigcap_i\ker\alpha_i \quad\text{as subspaces of }\mathfrak d.\] Since \(\mathfrak n\subseteq\mathfrak k\) and \(\mathfrak g=\mathfrak n+\mathfrak d\), inclusion gives \(\mathfrak g/\mathfrak k\cong\mathfrak d/(\mathfrak k\cap\mathfrak d)\), and passing through \(V\) proves (13). It also shows that every weight of \(\mathfrak n\) descends to \(W\). The statements about the integrated maps follow from simple connectivity. If \(\nu=0\), Lemma 5 gives nilpotence and \(W=0\). Conversely, if \(W=0\), all adjoint characters vanish, so every adjoint operator is strictly triangular and Engel’s theorem makes \(G\) nilpotent.

For \(d=\exp_D(v)\) and \(b_\gamma=\dim\mathfrak n_\gamma\), write \[\mathop{\mathrm{ad}}(v)|_{\mathfrak n_\gamma} =\gamma(\pi_*v)\mathop{\mathrm{id}}+T_\gamma(v), \qquad T_\gamma(v)^{b_\gamma}=0.\] The map \(v\mapsto T_\gamma(v)\) is linear, and exponentiation gives \[U_\gamma(d)= \sum_{j=0}^{b_\gamma-1}\frac{T_\gamma(v)^j}{j!}.\] Replacing \(v\) by \(-v\) gives its inverse. By (7), \(\|v\|\) is bounded by a polynomial in \(1+|d|_D\). The two finite sums therefore prove (14), uniformly over the finitely many weight spaces, including the zero weight when it occurs.

For \(v\in\mathfrak d\), the action of \(\mathop{\mathrm{ad}}v\) on \(\mathfrak g/\mathfrak n\cong \mathfrak d/(\mathfrak d\cap\mathfrak n)\) is nilpotent. Unimodularity of \(G\) consequently gives \[ \mathop{\mathrm{tr}}(\mathop{\mathrm{ad}}v|_{\mathfrak n}) =\mathop{\mathrm{tr}}(\mathop{\mathrm{ad}}v|_{\mathfrak g}) -\mathop{\mathrm{tr}}(\mathop{\mathrm{ad}}v|_{\mathfrak g/\mathfrak n})=0. \tag{17}\] Thus \(\det(\mathop{\mathrm{Ad}}_d|_{\mathfrak n})=1\). Bracket addition in (2) proves that \(S_w\) preserves brackets. Choosing \(v\in\mathfrak d\) over \(w\) in (17) also gives \[\det S_w =\exp\left(\sum_\gamma (\dim\mathfrak n_\gamma)\gamma(w)\right)=1.\] An automorphism of a simply connected nilpotent group changes Haar measure by its Lie algebra determinant, so both asserted actions preserve Haar measure. As \(N\) and \(D\) are unimodular and the action preserves Haar measure on \(N\), direct substitution in the semidirect product law proves that \(dn\,dd\) is left Haar measure on \(N\rtimes D\).

Finally, \(\pi_*\) annihilates \(\mathfrak d^2\), so in the splitting (5) it depends only on \(V_1\). The same projection is a homomorphism on the associated graded group. Equation (16) now follows from \(\delta_r|_{V_1}=r\mathop{\mathrm{id}}\). ◻

Proof. Write a word \(Y\in\mathcal Z\) in a maximizing basis as \(Y=\sum_{Z\in\mathcal B}c_{Y,Z}Z\). Whenever \(c_{Y,Z}\ne0\), replacing \(Z\) by \(Y\) gives another basis. Maximality gives \(h_Y(E)\le h_Z(E)\). Thus \[ e^{rh_Y(E)}Y= \sum_{Z\in\mathcal B}c_{Y,Z}e^{r(h_Y(E)-h_Z(E))} \big(e^{rh_Z(E)}Z\big) \tag{22}\] has uniformly bounded coefficients. There are only finitely many unscaled bases and words, so their coefficients admit a common bound. The reverse inclusion follows because the basis words belong to the list. This proves the first assertion.

The bracket of two basis words is either zero or a word of the list, with exponent equal to the sum of their exponents. Formula (22) therefore bounds every normalized structure constant. The Baker–Campbell–Hausdorff formula terminates at degree \(m\), which proves the assertion for a fixed number of factors. To control \(k\) factors, expand the formal product of their exponentials and then its logarithm, retaining only degrees at most \(m\). In degree \(j\) there are at most \(C_m(1+k)^j\) choices of factor indices, and each term contains at most \(j\) input coordinates. Converting these finitely many homogeneous Lie polynomials into brackets costs constants depending only on \(m\). The asserted polynomial bound in \(k,L\) follows from the bounded normalized structure constants.

Haar measure of a simply connected nilpotent group in exponential coordinates is a fixed multiple of Lebesgue measure. Indeed the Jacobian factor of the exponential map is \(\det((1-e^{-\mathop{\mathrm{ad}}v})/\mathop{\mathrm{ad}}v)=1\), interpreted by its finite power series, since \(\mathop{\mathrm{ad}}v\) is nilpotent. The determinant of the scaled basis is its unscaled determinant times \(e^{rP(E)}\). The finitely many unscaled determinants are bounded above and away from zero. This proves (20).

For an exact weight dilation, \(h_Z(E+w)=h_Z(E)+\gamma_Z(w)\). Every basis has total weight zero: it is a weight-homogeneous basis of \(\mathfrak n\), and the total weight is the trace character, which vanishes by Lemma 7. Hence maximizing bases remain maximizing after a translation of \(E\), and the claimed shift follows.

For conjugation, first remove the scalar action on each weight space. Lemma 7 bounds the remaining operator and its inverse polynomially in \(1+|d|_D\). Words of weight \(\alpha_i\) span the weight space containing \(X_i\), and each such word satisfies \[ h_Z(E)=\sum_{X_j\text{ in }Z}\max_E\alpha_j \ge\max_E\sum_{X_j\text{ in }Z}\alpha_j=s_i(E). \tag{23}\] Consequently the residual action on a scaled letter is a polynomially bounded combination of scaled words. Applying brackets and using the already established structure-constant bounds proves the same assertion for every scaled word. Restoring the character action is the exact shift by \(\pi d/r\). This gives (21); replacing \(d\) by \(d^{-1}\) proves its inverse version. ◻

Proof. Every basis from \(\mathcal Z\) contains all the letters \(X_i\): all longer words vanish in \(\mathfrak n/[\mathfrak n,\mathfrak n]\), and the letters project to its specified basis. As observed above, the sum of the word weights of any such basis is zero. Expanding this identity in letters gives \[ \sum_i c_i\alpha_i=0,\qquad c_i\text{ positive integers}. \tag{25}\] The total exponent of a basis is unchanged by translating \(E\), which proves translation invariance. The other two assertions about \(P\) follow from the corresponding properties of support functions and the finiteness of the list.

For a segment \([0,w]\), each support is \(\max(0,\alpha_i(w))\). These expressions give positive homogeneity, piecewise linearity, and the triangle inequality for \(p\), the last by taking the maximum of the basis sums after applying subadditivity term by term. Translation invariance gives \(p(-w)=p(w)\). If \(w\ne0\), the weights span \(W^*\), and (25) implies that some \(\alpha_i(w)>0\). All word exponents on the segment are nonnegative, and the mandatory letter \(X_i\) makes every basis sum positive. Thus \(p\) is a norm.

Every member of \(\mathcal A\) is a linear isometry of this norm. Its unit ball is a full-dimensional bounded polytope, so its linear isometry group injects into the permutation group of its finitely many vertices. In particular \(\mathcal A\) is finite.

Suppose \(A\in\mathcal A\) and \(A^*\alpha_j\) is not on one of the original oriented rays. From a Euclidean ball \(K\) centered at zero, remove a sufficiently small cap at the support point of \(A^*\alpha_j\). The resulting compact convex set \(E\) has the same supports in all original directions \(\alpha_i\), but a strictly smaller support in direction \(A^*\alpha_j\). Its supports in every other direction can only decrease. Since each candidate basis sum for \(P(AE)\) contains the letter \(j\), all these finitely many sums decrease strictly. Thus \(P(AE)<P(AK)=P(K)=P(E)\), contrary to \(A\in\mathcal A\). Pullback therefore maps the finite set of rays into itself; applying the same argument to \(A^{-1}\) proves that it permutes them. ◻

Proof. We construct a submersive scheme for each limiting normalized group. Its persistence under nearby group laws will then give uniform generation by compactness.

Pass first to a subsequence with the same maximizing basis. The normalized structure constants are bounded by Lemma 8, so a further subsequence converges. The Jacobi identity and the nilpotence-class bound pass to the limit, and its group law is the limiting BCH polynomial. Every letter is a basis vector. More generally, the bracket word giving a basis vector satisfies the exact identity \[Z(e^{rs_1(E)}X_1,\ldots,e^{rs_\nu(E)}X_\nu)=e^{rh_Z(E)}Z.\] Thus its normalized value is the same coordinate vector throughout the sequence and in its limit. The limiting letters therefore still bracket-generate the whole limiting algebra.

Here is a finite endpoint scheme with full rank in this limiting group. Let \(H\) be the subgroup generated by its letter one-parameter subgroups, and consider \[U=\mathop{\mathrm{span}}\{\mathop{\mathrm{Ad}}_hX_i:h\in H,\ 1\le i\le\nu\}.\] This space contains the letters and is invariant under \(\mathop{\mathrm{Ad}}_{\exp(tX_i)}\). Differentiating proves its invariance under \(\mathop{\mathrm{ad}}X_i\), so it contains all brackets and is the entire algebra. Choose \(d_N\) vectors \(\mathop{\mathrm{Ad}}_{h_j}X_{i_j}\) forming a basis, with each \(h_j\) a finite product of generator exponentials. The map \[ \Phi(t)=\prod_{j=1}^{d_N}h_j\exp(t_jX_{i_j})h_j^{-1} \tag{27}\] sends zero to the identity and has these basis vectors as its differential columns at zero. Expanding the \(h_j\) and their inverses gives a generator scheme as in (26). At a parameter point having fixed entries for the conjugating words and zero entries for the inserted increments, its differential is surjective. Enlarge a fixed parameter cube so this point is interior.

Use the same words and fixed parameters for the converging laws. Their endpoint maps converge with derivatives on compact parameter sets. The inverse function theorem, applied to the indicated \(d_N\) variables with the others fixed, supplies one fixed identity neighborhood contained in their images for all sufficiently late indices. Its local inverse supplies continuous parameter choices. For a fixed-radius log cube choose \(k\) so large that division of every log vector by \(k\) lies in this neighborhood. The identity \(\exp(v)=(\exp(v/k))^k\) covers the cube by a fixed number of copies of the scheme.

This proves the uniform generation assertion by contradiction: any sequence violating every proposed common length and parameter bound has a convergent normalized-law subsequence to which the preceding argument applies. Equivalently, the compact closure of the normalized laws has a finite cover by neighborhoods carrying the preceding inverse charts. There are only finitely many maximizing bases. One may pad words by identities, and concatenate the finitely many schemes, so there is no measurable choice among infinitely many alphabets. Local inverse choices on finite Borel partitions give the stated measurable endpoint choices.

For the growing-radius assertion let \(a_0=\min_i\|\alpha_i\|>0\). Enlarge \(E\) to \(E+\varepsilon_r\overline B_W(1)\), where \(\varepsilon_r=o(1)\) is large enough that \(r\varepsilon_ra_0\) dominates the logarithm of the radius and the fixed box-comparison constants. Every word exponent increases by at least \(\varepsilon_ra_0\). The all-word comparison then embeds the original growing box in a fixed-radius box for the enlarged template. Generation there changes each letter amplitude by only \(e^{o(r)}\).

Finally choose one sufficiently large centered ball as template. All its word exponents are strictly positive, so its boxes exhaust \(N\) as \(r\to\infty\). The uniform number of factors just established proves the last assertion. ◻

Proof. If \(\|\log_Nn\|\le e^r\), choose a fixed large centered ball \(E\) whose every word exponent is at least one. The finiteness of the unscaled bases gives \(n\in B_r(E,L_0)\) for a fixed \(L_0\). Lemma 10 writes it as a bounded product of letter factors with amplitudes at most \(Me^{rs_i(E)}\). Choose \(v_i\in\mathfrak d\) with \(\alpha_i(\pi v_i)=1\). Conjugating a letter factor back by \(d_i=\exp_D((rs_i(E)+b\log(2+r))v_i)\) removes its exponential amplitude. For a sufficiently large fixed \(b\), the extra logarithmic term also dominates the polynomial unipotent factor from Lemma 7. The conjugated-back log vector is bounded, while \(|d_i|_G=O(r+1)\). The resulting outward, bounded, and return moves prove the first estimate, also for bounded \(n\).

By left invariance the distance for a switch is the length of \(\mathop{\mathrm{Ad}}_{d^{-1}}\exp(tX_i)\). Its logarithm has norm at most \[C(1+r)^b|t|e^{-\alpha_i(\pi d)}.\] The first estimate makes this \(o(r)\) in the stated regime and \(O(\log(2+r))\) in the exact regime. ◻

Proof. Choose, for each letter, a support point \(v_i\in E_V\) for \(\alpha_i\circ\pi\). Fix a linear section \(V\to\mathfrak d\). From the pivot visit \(d_i=d_0\exp_D(r\widetilde{v_i-v_0})\), along the corresponding one-parameter segment. Its abelianization traces the segment \(r[v_0,v_i]\subset rE_V\), and its length is \(O(r)\). Decompose \(u\) using Lemma 10. To apply its next letter factor, move along this leg on the current \(D\)-chart, switch at \(d_i\), and reverse the leg on the new sheet. The support equality and Lemma 11 give the claimed switch cost. After the final return the endpoint is \(ud_0\).

Every continuous homomorphism \(l\) kills \(N\), since its differential kills \(\mathfrak n\subset[\mathfrak g,\mathfrak g]\), and factors on \(D\) through \(V\). It therefore has the specified values on the legs. It is Lipschitz for the fixed metrics, so each short switch changes its values by at most its Lipschitz constant times the switch length. This proves the third assertion.

Measurable generator choices are supplied by Lemma 10. Short connecting paths may also be chosen measurably. The Riemannian exponential map on a closed tangent ball of the asserted length radius is continuous from a compact space and covers the relevant metric ball. Such a map has a Borel section: use nested finite closed covers of successively smaller diameters and choose the first subpiece whose image contains the endpoint; all these image sets are compact, and the nested choices have one limiting point. The corresponding geodesics give the required choices. The same argument gives the subexponential version.

For the specialization to \(W\), choose an affine section of \(\pi:V\to W\) through \(a_D(d_0)/r\) over \(\pi d_0/r\), with one fixed linear part, and apply the construction to the image of \(E\). All leg displacements depend only on differences in that affine section. Their bounds therefore do not depend on its translating kernel component; the bound on \(|d_0|_D/r\) controls the positions. For a prescribed curve, replace the displayed legs by its portions between the pivot and a support position, and reverse each portion after the switch. All estimates remain the same. ◻

Proof. Translate the starting point to the identity and subdivide the path into at most \(C(1+r)\) increments of bounded length. Lemma 5 supplies bounded lifts \((v_j,e_j)\) of these increments. Their accumulated product has coordinates \[d_j=e_1\cdots e_j, \qquad u_j=\prod_{k=1}^j\mathop{\mathrm{Ad}}_{d_{k-1}}v_k.\] Thus \(|d_j|_D\le C(1+r)\), and \(\pi d_j\) is the actual height of the corresponding path vertex.

Expand \(\log_Nv_k\) in the fixed weight spaces. The polynomial residual action gives, in a weight \(\gamma\), coefficients bounded by \(C(1+r)^b e^{\gamma(\pi d_{k-1})}\). Words of that weight span its space, and \(\max_E\gamma\le h_Z(E)\) for every such word \(Z\). Consequently each conjugated increment has normalized coordinates bounded polynomially in \(r\). The product estimate in Lemma 8 gives the asserted bound for \(u_j\).

Bounded right thickening replaces \((u,d)\) by \((u\mathop{\mathrm{Ad}}_dv,de)\) with \((v,e)\) bounded. The same estimate at the terminal height controls this extra factor and proves the thickening assertion.

Equip \(N\rtimes D\) with a fixed left invariant Riemannian metric. The homomorphism \(q\) is globally Lipschitz: its derivative bound at the identity is transported everywhere by left translations. Therefore \(a\)-separated projected endpoints have separated chosen lifts, with separation bounded below by a fixed positive number. Small fixed-radius balls around those lifts are disjoint. The thickening assertion places all of them in \[B_r(E,C(1+r)^b)\times B_D(Cr+C).\] Product Haar is valid by Lemma 7; the first factor has volume at most a polynomial times \(e^{rP(E)}\), and the second has polynomial volume by Lemma 6. Every small lifted ball has the same positive Haar volume. This proves (28). Injectivity of \(q\) is not used.

For the sequential statement, enlarge \(E\) by a vanishing-radius ball so it contains the actual relative path heights divided by \(r\) and zero. The support functional changes by \(o(1)\), and the box exponents by \(o(1)\). Polynomial factors are subexponential. For an unrestricted path of length \(t\), its intermediate heights are \(O(1+t)\) because \(\pi\) is Lipschitz. The same lifted-increment argument now bounds each coordinate by \(e^{O(1+t)}\), and its polynomially many BCH terms preserve this bound. ◻

Proof. If \(N=1\), then \(G=D\) is nilpotent. The endpoints lie in the radius-\(C_0t\) ball, and disjoint fixed-radius balls about separated points, together with Lemma 6, give the polynomial bound directly. Suppose now that \(N\ne1\). Repeat the bounded-increment lifting in the preceding proof. Every accumulated \(D\)-coordinate has intrinsic length \(O(1+t)\) and height norm at most \(M\). On each of the finitely many weight spaces the character factor is bounded by a constant depending on \(M\), while the residual action is polynomial in \(1+t\). Hence every conjugated \(N\)-increment has polynomial log norm. There are \(O(1+t)\) increments; the fixed-degree nilpotent product formula bounds the log norm of their product by \(C(1+t)^b\). The same bounds survive fixed bounded right thickening. The allowed \(N\) log-coordinate region and the intrinsic \(D\)-ball both have polynomial Haar volume. Disjoint small balls around lifts of separated endpoints give the result exactly as before. In particular this is a finite polynomial bound; it does not follow by interpreting a subexponential error as polynomial. ◻

Proof. Set \(K_a=N\cap\overline B_G(1,a)\). This is compact since \(N\) is closed and the metric on \(G\) is proper. If \(nd\) and \(n_0d\) have distance at most \(a\), then \(n_0^{-1}n\in\mathop{\mathrm{Ad}}_dK_a\). Conjugation by \(d\) preserves \(N\)-Haar, so the possible \(n\) have measure at most \(\mathop{\mathrm{vol}}_N K_a\), independently of \(d\). A maximal \(a\)-separated subset of the bounded slice covers it by radius-\(a\) balls. Summing the preceding measure bound gives the assertion, after a harmless increase of \(a\) or of \(C_a\) at boundary points. Left translation by \(x\) is an isometry. In the product maximum metric the covering balls are products, so the same proof multiplies the constants. ◻

Proof. If the set is nonempty, fix one representative \((u_0,d_0)\). Every other projected point differs from it by an element of a fixed radius-\(2R\) ball. Choose a bounded lift \((v,e)\) of that difference, using Lemma 5. In \(G\), \[ud=u_0d_0ve=u_0\mathop{\mathrm{Ad}}_{d_0}(v)d_0e.\] It follows that \[ u_0^{-1}u=\mathop{\mathrm{Ad}}_{d_0}(v)a, \qquad a=d_0ed^{-1}\in N\cap D, \qquad |a|_D\le2T+O_R(1). \tag{29}\] The first factor has log norm at most \(C(1+T)^b e^{cH}\) by the character and unipotent bounds. The second has polynomial log norm by intrinsic nilpotent coordinates in \(D\). The logarithm of \(a\) in \(D\) and in \(N\) is the same vector in \(\mathfrak d\cap\mathfrak n\), because both are restrictions of the injective exponential map of \(G\). The finite BCH formula therefore places \(u_0^{-1}u\) in an \(N\) log-coordinate ball of radius \(C(1+T)^{b'}e^{c'H}\). Its Lebesgue, hence Haar, volume has the stated form after changing the constants. This proves the bound even for outer measure and uses no centrality of the overlap. ◻

Proof. We first keep a definite part of \(E\) away from the zero set of the chosen derivative minor. A coordinate slice then retains positive measure, and a small cube in that slice makes the restricted map injective. Change of variables on this cube will give the image-volume bound.

The first step uses a scalar sublevel estimate. If a polynomial \(Q\) in \(k\ge1\) variables has degree at most \(d\ge1\) and some coefficient of absolute value at least \(c\), then, for \(R\ge1\) and \(0<\eta<c\), \[ \big|\{x\in[-R,R]^k:|Q(x)|\le\eta\}\big| \le C_{k,d}(1+R)^k(\eta/c)^{1/(kd)}. \tag{32}\] No upper coefficient bound is needed here.

For \(k=1\), let the sublevel set have length \(l>0\). It contains \(d+1\) points whose mutual distances are at least \(l/[2(d+1)]\): after fewer than \(d+1\) choices, deleting intervals of that radius about the chosen points removes total length strictly less than \(l\). Lagrange interpolation at these points bounds every coefficient of \(Q\) by \[C_d\eta(1+R)^d l^{-d}.\] Indeed the denominators are products of \(d\) point differences, while the coefficients of the numerator polynomials are bounded by \(C_d(1+R)^d\). The asserted lower bound on one coefficient now proves (32) in one variable.

For the induction step, write \[Q(x',t)=\sum_{j=0}^d q_j(x')t^j\] and choose \(q_j\) which contains a coefficient of absolute value at least \(c\). Put \(u=\eta/c\in(0,1)\) and \(\tau=c u^{(k-1)/k}\). The induction hypothesis bounds the \((k-1)\)-dimensional measure of \(\{|q_j|\le\tau\}\) in the base cube by \[C(1+R)^{k-1}u^{1/(kd)}.\] Above its complement, the one-variable estimate bounds the length of each sublevel slice by \[C(1+R)(\eta/\tau)^{1/d} =C(1+R)u^{1/(kd)}.\] Fubini’s theorem, using the bound \(2R\) on the remaining slice lengths, proves (32). The argument also applies to constant polynomials, since \(d\) is only a degree upper bound.

Return to the minor \(Q\) of \(D\Phi\) in the statement. Its degree is at most \(b(D-1)\), so we can use \(d=\max\{1,b(D-1)\}\). Choose a compact \(K\subset E\) with \(|K|\ge m/2\). For a sufficiently large constant \(C_0\), depending only on \(a,b,D\), set \[\eta=c_0\left(\frac{m}{C_0(1+R)^a}\right)^{ad}.\] Since \(m\le(2R)^a\), we have \(0<\eta<c_0\le1\) after increasing \(C_0\). Estimate (32), with coefficient lower bound \(c_0\), makes the bad-minor set have measure at most \(m/4\). Therefore the compact set \[K'=K\cap\{|Q|\ge\eta\}\] has measure at least \(m/4\).

Permute the domain coordinates so that the chosen minor uses the first \(b\) columns, and write \(x=(v,z)\) with \(v\in\mathbb R^b\). Fubini supplies a fixed \(z\) for which the compact slice \(K'_z\) satisfies \[ |K'_z|\ge C_a m(1+R)^{-(a-b)}. \tag{33}\] When \(a=b\), there is no slicing and we use \(K'\) itself. For \(f(v)=\Phi(v,z)\), its derivative and the Lipschitz constant of its derivative on \([-R,R]^b\) are both at most \[L=C_{a,b,D}(1+B)(1+R)^D\ge1.\]

We now retain a small cube on which \(f\) is injective. Cover \([-R,R]^b\) by at most \(C_b((1+R)/\ell)^b\) closed cubes of side at most \[\ell=\frac{\eta}{4\sqrt b\,L^b}.\] One such cube \(J\) contains at least the reciprocal of this number times the mass in (33). Choose \(v_0\in K'_z\cap J\). The least singular value of \(Df(v_0)\) is at least \(\eta/L^{b-1}\), since its determinant has absolute value at least \(\eta\). For every \(v\in J\), \[\|Df(v)-Df(v_0)\| \le L\sqrt b\,\ell \le\frac{\eta}{4L^{b-1}}.\] Integrating along a line segment in \(J\) gives \[|f(v)-f(w)|\ge\frac{3\eta}{4L^{b-1}}|v-w| \qquad(v,w\in J).\] Thus \(f\) is injective on \(J\). Change of variables on its interior, with the boundary omitted as a null set, gives \[|f(K'_z\cap J)| =\int_{K'_z\cap J}|Q(v,z)|\,dv \ge\eta |K'_z\cap J|.\] The image here is compact and is contained in \(\Phi(E)\). Combining the cube count and (33) yields \[|\Phi(E)|_{\mathrm{in}} \ge C m\eta^{b+1}(1+R)^{-a}L^{-b^2}.\] Substituting \(\eta\) and \(L\) proves (30) with the stated exponents. This proof uses neither finite global fibers nor a global multiplicity bound.

Finally, a polynomial map submersive somewhere has a derivative minor which is a nonzero polynomial. Coefficient convergence gives a fixed positive lower bound for one coefficient of that same minor of \(\Phi_r\), for all sufficiently large \(r\), and a fixed upper bound for all coefficients of \(\Phi_r\). Every factor in (30) then has logarithm bounded below by \(-o(r)\) under the hypotheses on \(R_r\) and \(|E_r|\). The final variant follows from the same displayed estimate. ◻

Proof. For a Borel set \(A\subset Q\), let \[(T_j1_A)(x) =\int\!\int 1_A(x s_1s_2^{-1}) \,d\sigma_j(s_1)\,d\sigma_j(s_2).\] This is a Borel function. Right Haar invariance, the substitution \(z=x s_1\), and Fubini give \[\begin{align*} \int_A T_j1_A(x)\,d\mu(x) &=\int_J\left(\int 1_A(zs^{-1})\,d\sigma_j(s)\right)^2d\mu(z)\\ &\ge\frac{\mu(A)^2}{\mu(Q\mathop{\mathrm{supp}}\sigma_j)} \ge\frac{\mu(A)^2}{D_j\mu(Q)}. \end{align*}\] For the first inequality, the inner function is supported in \(Q\mathop{\mathrm{supp}}\sigma_j\), has integral \(\mu(A)\), and Cauchy–Schwarz applies. Compactness makes the containing set measurable and of finite measure.

Starting with \(S_h=S\), work backwards and define \[S_{j-1} =\{x\in S_j:(T_j1_{S_j})(x)\ge\tau_j\}, \qquad \tau_j=\frac{\mu(S_j)}{2D_j\mu(Q)}.\] The contribution from \(S_j\setminus S_{j-1}\) to the preceding integral is at most \(\mu(S_j)^2/(2D_j\mu(Q))\). Since \(T_j1_{S_j}\le1\), the remaining contribution implies \[\mu(S_{j-1})\ge\frac{\mu(S_j)^2}{2D_j\mu(Q)},\] which proves (34); the definition gives (35). Conditional on the first \(j-1\) successful transitions, the current state lies in \(S_{j-1}\) and the next increment retains its prescribed law. Multiplying the conditional lower bounds proves the claimed probability estimate. For fixed \(h\), finitely many squarings, products, and divisions by \(e^{o(r)}\) preserve a lower bound of the form \(e^{-o(r)}\). ◻

Proof. Apply Lemma 18 and choose \(x_r\in S_{0,r}\). Independent sampling of the parameters produces the required independent increment laws, so \(\nu_r(E_r)\ge e^{-o(r)}\). The set \(E_r\) is Borel, because \(\Theta_r\), multiplication, and the increment maps are Borel. The density upper bound gives \[\nu_r(E_r)\le e^{o(r)}|E_r|,\] proving the first conclusion. Lemma 17 proves the second. No independence remains necessary after restricting to \(E_r\). ◻

Proof. On \(X\), the composition bound gives \(d(\bar F(e),e)\le\Lambda\), since \(F(e)=e\). On \(Y\) it similarly gives \(d(F(e),e)\le\Lambda\). Equi-Lipschitz bounds now give pointwise precompactness on every compact subset of \(G\). Arzelà–Ascoli and a countable compact exhaustion yield compactness; all the inequalities and composition bounds are closed under compact-open convergence. The same exhaustion metrizes the topology.

The inverse of the first coordinate map is \[\omega\longmapsto \bigl(F_\omega(e),L_{F_\omega(e)^{-1}}\omega\bigr),\] and that of the second is \[\omega\longmapsto \bigl(\omega\cdot\bar F_\omega(e),\bar F_\omega(e)^{-1}\bigr).\] They are continuous. Continuity of evaluation and composition follows directly from compact-open convergence and the common Lipschitz bound. The action identities follow either from these coordinates or by substituting the two normalization formulas. For example, \[F_{T_x\xi}(z)=F_\xi(x)^{-1}F_\xi(xz),\qquad \bar F_{U_y\eta}(z)=\bar F_\eta(y)^{-1}\bar F_\eta(yz).\] These identities also verify the stated right-action convention. ◻

Proof. The representation \(\rho\) is unitary because \(\mu\) is invariant. It is strongly continuous: this follows first on continuous functions from the continuous action on compact \(X\), and then on \(L^2\) by density and unitarity. The jointly continuous evaluation \((x,\xi)\mapsto\pi F_\xi(x)\) is uniformly continuous on compact subsets of \(G\times X\), so \(b\) is a continuous \(L^2\) cocycle.

Real-triangulable Lie groups have property \(\mathcal{WAP}_{\mathrm t}\), and hence property \(H_{\mathrm t}\), by (Cornulier and Tessera 2020, Corollary 1.9). In the form needed here, every continuous unitary representation without invariant vectors has zero first reduced cohomology. Reduced cohomology uses the closure of the coboundaries for uniform norm convergence on compact subsets of the group. The Hilbert representations here are weakly almost periodic, so the cited theorem applies.

Ergodicity identifies the invariant vectors in \(L^2(X,\mu;W)\) with the constant functions. Orthogonal projection of (36) onto this subspace gives the continuous homomorphism \[l(x)=\int_X\pi F_\xi(x)\,d\mu(\xi).\] Apply the vanishing theorem to the cocycle \(b-l\) in the orthogonal complement. It gives (37). One may complexify the real Hilbert space and take real parts if using the complex convention for unitary representations.

For completeness, fix a compact symmetric generating neighborhood \(Q\) such that every \(x\) is a product of at most \(a|x|_G+a\) elements of \(Q\). A fixed-radius metric ball has this property by subdividing a geodesic. Set \[c_j(x)=b(x)+\rho(x)\phi_j-\phi_j-l(x).\] Given \(\epsilon>0\), fix \(j\) with \(\sup_{x\in Q}\|c_j(x)\|_2\le\epsilon\). The cocycle identity and unitarity imply \[\|b(x)-l(x)\|_2 \le (a|x|_G+a)\epsilon+2\|\phi_j\|_2.\] Keep this \(j\) fixed, let \(r\to\infty\) for \(|x|_G\le Rr\), and then let \(\epsilon\downarrow0\). This proves (38). It is a supremum of \(L^2\) norms, which will be sufficient for finite meshes of paths.

A continuous homomorphism to the additive group \(W\) kills the derived Lie algebra. Since \(\mathfrak n\subset[\mathfrak g,\mathfrak g]\) and \(N\) is connected, it kills \(N\); on \(D\) it factors through the abelianization \(V\). ◻

Proof. Fix \(v_0\in E_V\) and a pivot \(d_0\in D\) of intrinsic length \(O(r)\) whose abelianization is \(rv_0\). Put \(E=\pi E_V\). Apply Lemma 12 with the set \(E_V\subset V\): although \(l\) factors through \(V\) on \(D\), we do not yet know that it factors through \(W\). For each \(n\in B_r(E)\), that lemma supplies a measurably chosen path from \(d_0\) to \(nd_0\) of length \(O(r)\). Its abelianized \(D\)-legs stay in \(rE_V\), and its switches have length \(o(r)\). Since \(l\) kills \(N\) and is Lipschitz, the path’s \(l\)-heights lie in \(r(lE_V+o(1))\). All its points have distance \(O(r)\) from \(e\), with common bounds for the sampled endpoints.

Sample \(n\) with normalized Haar measure on \(B_r(E)\), independently of \(\xi\sim\sigma\). Fix \(\tau>0\) and sample each path at an arclength mesh of spacing at most \(\tau r\), including its endpoints. There are at most \(M_\tau\) sampled points, independent of \(r,n\). At any of them, say \(g_r(n)\), the assumed estimate gives \[\int_X|\pi F_\xi(g_r(n))-l(g_r(n))|^2\,d\sigma(\xi)=o(r^2),\] uniformly in \(n\). Fubini, a finite union bound, and Chebyshev’s inequality show that, for every fixed \(\eta>0\), the fraction of pairs \((\xi,n)\) on which a sampled error exceeds \(\eta r\) tends to zero. Hence for all large \(r\) there is one \(\xi\) for which at least half the endpoint volume has no such error.

For these endpoints the entire image path lies at heights within \((c\tau+\eta+o(1))r\) of \(r lE_V\). Indeed the image of each mesh interval has diameter at most the common Lipschitz constant times \(\tau r\). The initial image point is the common \(F_\xi(d_0)\); subtracting its height only translates the allowed height set.

At their common source \(D\)-position, the good endpoints have a separated subset of cardinality at least \(c_0 e^{rP(E)}\), by Lemma 15. Choose the separation large enough that the quasi-isometry lower bound keeps their images separated by a fixed positive amount. Their image paths have length \(O(r)\). Lemma 13 and translation invariance of \(P\) give \[P(E)\le P\bigl(lE_V+\overline B_W(0,c\tau+\eta)\bigr)\] after taking \(r\to\infty\). First let \(\eta\downarrow0\), then \(\tau\downarrow0\). Continuity of \(P\) for Hausdorff distance proves (39). This argument only used a fixed finite mesh before the limit; it did not require an \(L^2\) bound on the pointwise supremum over a path.

Apply (39) to \([0,v]\subset V\). Since \(p\) is a norm, \[p(lv)\ge p(\pi v),\qquad \ker l\subseteq\ker\pi.\] Writing \(d=\dim W\), the inclusion forces \(\mathop{\mathrm{rank}}l\ge d\), while the target of \(l\) has dimension \(d\). Thus \(l\) and \(\pi\) have the same kernel on \(V\), and \(l=A\pi\) there for an invertible \(A\). Both homomorphisms kill \(N\) and \(G=ND\), so this equality holds on \(G\). All the reasoning applies to the reversed pairs on \(Y\) as well. ◻

Proof. The invariant measure on the inverse section. Let \(m\) be Haar measure on \(G\). In the coordinates \(\Omega=G\times X\) of Lemma 22, put \(M=m\times\mu\). This is a nonzero Radon measure and is target-invariant. Domain translation has the formula \[(z,\xi)\cdot x=(zF_\xi(x),T_x\xi).\] For each fixed \(\xi\), the change in \(z\) is right translation, which preserves \(m\) by unimodularity. Invariance of \(\mu\) therefore makes \(M\) domain-invariant as well.

In the other coordinates \(\Omega=Y\times G\), domain translation is \((\eta,x)\mapsto(\eta,xg)\). For a Borel set \(E\subset Y\), the measure \(B\mapsto M(E\times B)\) is right-invariant and locally finite: for compact \(B\) it is bounded by \(M(Y\times B)<\infty\). Haar uniqueness gives \(M(E\times B)=\nu(E)m(B)\), and countable additivity in \(E\) supplies a Borel measure \(\nu\) with \[ M=\nu\times m\quad\hbox{on }Y\times G. \tag{42}\] Here a product with compact factors is compact in \(\Omega\), by the coordinate homeomorphism, so the local finiteness used above follows from the Radon property of \(M\).

Choose a compact ball \(B_0\subset G\) with \(0<m(B_0)<\infty\). Then \[\nu(Y)m(B_0)=M(Y\times B_0)<\infty.\] If \(\nu(Y)=0\), a countable compact exhaustion of \(G\) and (42) would imply \(M=0\). Thus \(0<\nu(Y)<\infty\). In particular, \(\nu\) is a measure obtained from product coordinates; we are not restricting \(M\) to the possibly \(M\)-null section \(Y\).

For \(y\in G\), set \(q_y(\eta)=\bar F_\eta(y)\). The identity \[ L_{y^{-1}}(\eta\cdot x) =(U_y\eta)\cdot(q_y(\eta)^{-1}x) \tag{43}\] shows that \(\nu\) is \(U\)-invariant. More explicitly, integrate a test function of \(\eta\) times a compactly supported function of \(x\) against \(M\). In the transformed integral, the fiber change \(x\mapsto q_y(\eta)^{-1}x\) preserves left Haar measure. Target invariance of \(M\) then gives invariance of the \(\eta\)-integral.

Transfer functions averaged over the fibers. Choose measurable representatives of the \(\phi_j\) and lift them by \[\Phi_j(L_z\xi)=\pi z+\phi_j(\xi).\] They belong to \(L^2\) on every compact subset of \(\Omega\) and satisfy \(\Phi_j(L_z\omega)=\pi z+\Phi_j(\omega)\). Their domain increments have defect \[E_j(\omega,t)=\Phi_j(\omega\cdot t)-\Phi_j(\omega)-A\pi t.\] If \(\omega=L_z\xi\), this is exactly \(c_j(t)(\xi)=b(t)(\xi)+\phi_j(T_t\xi)-\phi_j(\xi)-A\pi t\). Consequently, for compact \(K_\Omega\subset\Omega\) and \(K_G\subset G\), \[ \sup_{t\in K_G}\int_{K_\Omega}|E_j(\omega,t)|^2\,dM(\omega) \longrightarrow0. \tag{44}\] Indeed \(K_\Omega\) lies in \(K\times X\) for a compact \(K\subset G\) in the first coordinates; the integral is at most \(m(K)\|c_j(t)\|_2^2\).

We now average over whole balls so that no trace of an \(L^2\) function on a null section is required. Fix a compact ball \(B\subset G\) of positive Haar measure, and define \[\begin{align*} D_j(\eta,x)&=\Phi_j(\eta\cdot x)-A\pi x,\\ \Psi_j(\eta)&=\frac1{m(B)}\int_B D_j(\eta,x)\,dm(x). \end{align*}\] Product coordinates and Fubini define this average for \(\nu\)-almost every \(\eta\). Jensen’s inequality and local \(L^2\) integrability of \(\Phi_j\) show that \(\Psi_j\in L^2(Y,\nu;W)\); define it arbitrarily on the remaining null set.

Equation (43), target equivariance of \(\Phi_j\), and additivity of \(\pi\) give \[ \Psi_j(U_y\eta)-\Psi_j(\eta) =-\pi y+A\pi q_y(\eta)+R_j(\eta,y), \tag{45}\] where \(R_j(\eta,y)\) is the difference of the averages of \(D_j(\eta,\cdot)\) on \(q_y(\eta)B\) and on \(B\). For each fixed \(y\), this identity holds almost everywhere: \(U_y\) preserves \(\nu\), and the changes in the fiber variable preserve Haar measure. These facts also show that changing the measurable representatives of the transfers changes none of their \(L^2\) classes or averages.

Comparing two fiber averages. We claim that \(R_j\) tends to zero in \(L^2(\nu)\) uniformly for \(y\) in any fixed compact \(Q\). The increment estimate (44) will control, in \(L^2(\nu)\), the difference between averages over two bounded regions of each fiber. Since \(\bar F_\eta(e)=e\) on \(Y\) and the maps have a common Lipschitz constant, all \(q_y(\eta)\) with \(y\in Q\), \(\eta\in Y\) lie in one compact set \(Q_1\subset G\). Put \(C_1=Q_1B\). Jensen’s inequality applied to the two averages gives, for every \(y\in Q\), \[\begin{align*} \|R_j(\cdot,y)\|_2^2 &\le \frac1{m(B)^2} \int_Y\int_{C_1}\int_B |D_j(\eta,x_1)-D_j(\eta,x_2)|^2\, dm(x_2)\,dm(x_1)\,d\nu(\eta). \tag{46}\end{align*}\] The enlargement from \(q_y(\eta)B\) to \(C_1\) is legitimate because its averaging density is bounded by \(1/m(B)\), even though its center depends on \(\eta\). Thus the right side is independent of \(y\).

Inside this integral one has \[D_j(\eta,x_1)-D_j(\eta,x_2) =E_j(\eta\cdot x_2,x_2^{-1}x_1).\] Make the substitution \(t=x_2^{-1}x_1\), which preserves left Haar measure in \(x_1\). Enlarge the resulting \(t\)-domain to the compact \(B^{-1}C_1\). The pairs \(\eta\cdot x_2\) range over the compact image of \(Y\times B\), and their measure is \(M\) by (42). Hence the right side of (46) is at most \[\frac1{m(B)^2}\int_{B^{-1}C_1} \int_{Y\cdot B}|E_j(\omega,t)|^2\,dM(\omega)\,dm(t),\] which tends to zero by (44). We have proved \[\sup_{y\in Q}\|R_j(\cdot,y)\|_{L^2(\nu)}\longrightarrow0.\] The supremum here is over the individual \(L^2\) norms. No common pointwise exceptional set for all \(y\) is being asserted or used.

Set \(\theta_j=-A^{-1}\Psi_j\) and rearrange (45). This gives \[\pi\bar F_\eta(y)+\theta_j(U_y\eta)-\theta_j(\eta) =A^{-1}\pi y-A^{-1}R_j(\eta,y),\] proving (40) with precisely the inverse of the given \(A\). The inverse height evaluations form a continuous cocycle for the unitary action on \(L^2(Y,\nu;W)\), by the formulas in Lemma 22. Subdividing a geodesic into bounded steps, exactly as in the proof of Lemma 23, proves (41). Neither ergodicity of \(\nu\) nor a second application of reduced cohomology is needed. ◻

Proof. Lemma 23 gives \(l\) and its compact coboundary approximations. Lemma 24 gives \(l=A\pi\) with \(A\) invertible. The inverse coupling in Lemma 25 supplies a finite nonzero invariant measure \(\nu\) and the inverse averaged estimate with the same \(A^{-1}\). Normalize \(\nu\) to a probability and apply Lemma 24 to the inverse maps.

Every compact convex \(E\subset W\) has a compact convex lift to \(V\), by a fixed linear section of \(\pi:V\to W\). The forward comparison therefore gives \(P(E)\le P(AE)\), and the inverse comparison applied to \(AE\) gives \(P(AE)\le P(E)\). Thus \(P(AE)=P(E)\) for every nonempty compact convex \(E\), which is precisely \(A\in\mathcal A\). ◻

Proof. Write \(\mu=\int\mu_s\,d\kappa(s)\) for its ergodic decomposition (Greschonig and Schmidt 2000, Theorem 5.2). On almost every component Proposition 26 gives \(A_s\in\mathcal A\). Since \(|\exp_D(Lv)|_G\le L\|v\|\) up to the fixed norm convention, its estimate implies convergence of the displayed integral with \(\mu_s\) in place of \(\mu\).

The normalized height evaluations are uniformly bounded for \(L\ge1\) by the Lipschitz bounds for \(F\) and \(\pi\) and by \(F(e)=e\). The comparison set is finite and fixed. Dominated convergence through the ergodic decomposition proves (47). No measurable choice of \(A_s\) is required: the integrand already takes distance to the full finite set. Continuity for fixed \(L\) follows from evaluation in the compact-open topology. ◻

Proof. Expand \(\log k_0\) in the finite list of bracket words using Lemma 8. Its coefficient in a word \(Y\) is bounded by \(\exp(r(1+\delta)h_Y(\mathcal C)+o(r))\). The finite expansion of \(\exp(\mathop{\mathrm{ad}}(\log k_0))\log u_r\) consists of brackets of an initial word \(Z\) with additional words \(Y_1,\ldots,Y_j\). The sum of their lengths is at most \(m\) whenever the bracket is nonzero. Before expansion in a maximizing basis, the logarithm of the coefficient divided by \(r\) is at most \[(1-c\delta)h_Z(\mathcal C) +(1+\delta)\sum_{\ell=1}^j h_{Y_\ell}(\mathcal C)+o(1).\] The width of the resulting word is \(h_Z(\mathcal C)+\sum_\ell h_{Y_\ell}(\mathcal C)\). Relative to this width the exponent is therefore at most \[ -\delta\left(c h_Z(\mathcal C) -\sum_{\ell=1}^j h_{Y_\ell}(\mathcal C)\right)+o(1) \le -\delta(cs_--ms_+)+o(1). \tag{53}\] This is bounded above by a negative constant for all large \(r\). The basis-exchange comparison in Lemma 8 says that expanding a word in a maximizing basis uses only basis entries whose widths are at least its width. That expansion cannot increase the bound in (53). There are only finitely many terms, which proves the assertion. ◻

Proof. We prove that the laws of the sampled points are asymptotically unchanged in total variation by right multiplication by each fixed element of \(G\). It suffices to consider elements of \(D\) and \(N\), since these subgroups generate \(G\).

For \(d\in D\), right multiplication changes \(z\) to \[z\longmapsto\Delta_r^{-1}(\Delta_r z\,d).\] This map preserves Haar measure in the scaled coordinates, because \(D\) is unimodular and the coordinate scaling has constant Jacobian. By Lemma 6 it converges to the identity uniformly on compact sets. A continuous compactly supported density changes by \(o(1)\) in \(L^1\), proving the assertion for \(d\).

For fixed \(u\in N\), right multiplication changes \(a\) to \[a\,q_r(z),\qquad q_r(z)=k_0\mathop{\mathrm{Ad}}_{\Delta_r z}(u)k_0^{-1},\] and leaves \(z\) unchanged. On the compact support of \(\rho\), the weight-action estimate in Lemma 7 bounds a coefficient of \(\log\mathop{\mathrm{Ad}}_{\Delta_r z}(u)\) in a word \(Z\) by \[\exp\bigl(r\gamma_Z(\pi z)+o(r)\bigr) \le \exp\bigl(r(1-c\delta)h_Z(\mathcal C)+o(r)\bigr).\] The error is uniform there; it includes the polynomial factors from the unipotent part of the \(D\) action. Words of zero total weight cause no exception to the inequality. Lemma 29 shows that \(q_r(z)\) tends exponentially to the identity in the \(\mathcal C\)-normalized coordinates, uniformly in \(z\).

The normalized product laws have uniformly bounded polynomial coefficients. Thus, on the fixed coordinate cube defining \(B_r(\mathcal C)\), multiplication by \(q_r(z)\) and by \(q_r(z)^{-1}\) both converge uniformly to the identity. Their images lie in Euclidean boundary enlargements of the cube of thickness tending to zero. Haar invariance now gives \[\sup_{z\in\mathop{\mathrm{supp}}\rho} \frac{\mathop{\mathrm{vol}}_N(B_r(\mathcal C)q_r(z) \mathbin{\triangle} B_r(\mathcal C))} {\mathop{\mathrm{vol}}_N(B_r(\mathcal C))}\longrightarrow0.\] This is precisely the required total-variation estimate for the conditional uniform laws of \(a\).

The estimates were proved before projecting the sampling coordinates to \(G\). Projection decreases total variation, so the possible nonuniqueness of \(ND\) coordinates is harmless here. Finally apply (54). The action on the compact space \(X\) is continuous by Lemma 22; every weak limit of the normalized occupation laws is therefore invariant. ◻

Proof. First allow the starting point \(z\) to vary in a small neighborhood \(O\) so that the whole family of segments lies in a compact subset of \(\mathcal U_\delta\). Choose a continuous probability density \(\eta\) supported in \(O\), positive near the specified starting point. Choose another continuous probability density \(\rho\), compactly supported in \(\mathcal U_\delta\), which is bounded below by a positive number on a compact neighborhood of this swept family. The same neighborhood contains all the actual scaled paths \(\Psi_{r,t}(z)\) from (52), for all large \(r\), \(z\in\mathop{\mathrm{supp}}\eta\), and \(0\le t\le T\).

Let \(\mu_r\) be the occupation law obtained with \(\rho\). Pass to a subsequence on which \(\mu_r\) converges to \(\mu\); the prescribed profile limit remains \(\nu\). By Lemma 30, \(\mu\) is invariant. The cutoff and this subsequence are fixed before choosing a microstep length.

Fix an integer \(L\ge1\). Put \(q=\lfloor rT/L\rfloor\) and divide the actual path into \(q\) increments by \(\exp_D(Lv)\) and a final remainder. For \(0\le j<q\), its \(j\)th starting point is \[p_j=xak_0\Delta_r z\exp_D(jLv).\] Right invariance of Haar measure in \(D\) shows that the distribution of the scaled \(D\) coordinate of \(p_j\), when \(z\) has density \(\eta\), has density at most \(\|\eta\|_\infty\). Its support is in the compact set where \(\rho\) has a positive lower bound. Hence the law of \(\omega[p_j]\) is bounded above by \(C_0\mu_r\), with a constant \(C_0\) independent of \(r,j,L\). The original law of \(a\) is unchanged and independent of \(z\) throughout this comparison.

Write \(K_v=\operatorname{conv}\{A\pi v:A\in\mathcal A\}\) and define the bounded continuous function on \(X\) \[d_L(\omega')= \mathop{\mathrm{dist}}\left(\frac{\pi F_{\omega'}(\exp_D(Lv))}{L},K_v\right).\] The identity of height cocycles at \(p_j\) and convexity give \[\begin{align*} &\mathbb E_{a,z}\mathop{\mathrm{dist}}\left( \frac{f_{r,x,ak_0}(\Psi_{r,T}(z))-f_{r,x,ak_0}(z)}{T},K_v\right) \\ &\hspace{25mm}\le \frac{L}{rT}\sum_{j=0}^{q-1}\mathbb E_{a,z}d_L(\omega[p_j]) +O(L/r) \le C_0\int_X d_L\,d\mu_r+O(L/r). \end{align*}\] The final error uses the Lipschitz bound on the remainder interval and the boundedness of \(K_v\). For this fixed \(L\), let \(r\to\infty\). Profile compactness and the uniform convergence of \(\Psi_{r,T}\) give \[\int\!\int_O \mathop{\mathrm{dist}}\left(\frac{f'(z\exp_\infty(Tv))-f'(z)}{T},K_v\right) \eta(z)\,dz\,d\nu(f') \le C_0\int_X d_L\,d\mu.\] Now let the integer \(L\) tend to infinity. Corollary 27 for this one fixed invariant probability \(\mu\) makes the right side tend to zero. No convergence rate uniform over invariant probabilities is required.

The nonnegative integrand on the left is continuous in the profile and the starting point. Its integral being zero, it vanishes on the product of the supports. In particular it vanishes for the specified \(f\) and \(z\). The argument applies to each segment as stated in the lemma. ◻

Proof. Fixing the endpoint scheme. Set \(E=\pi J\). We may pass to any further subsequence of the one giving \(\nu\). First pass to a subsequence on which the nilpotent laws in \(E\)-normalized coordinates converge. By Lemma 10, choose a fixed finite ordered list of generator increments whose normalized endpoint map has a submersive limit at an interior parameter point. Its length \(h\) is fixed independently of \(r\). An increment in letter \(i\) has the form \(\exp(t e^{rs_i(E)}X_i)\), with \(t\) in a fixed bounded interval. Replace \(t\) by \(t_1-t_2\) with independent uniform interval parameters, enlarging their fixed intervals if necessary. The difference map is surjective, so the limiting endpoint map is still submersive somewhere in the parameter cube. Fix this scheme now.

Retaining sheets with the prescribed profile. Choose a compact neighborhood \(Z_J\Subset\mathcal U_\delta\) of \(J\) that contains all the actual scaled paths \(\Psi_{r,t}(z)\), \(0\le t\le T\), for large \(r\). Put \(Q_r=B_r(\mathcal C)k_0\). Because \(f\in\mathop{\mathrm{supp}}\nu\), for each fixed positive tolerance the set of \(n\in Q_r\) whose profiles are within that tolerance of \(f\) on this compact has a positive lower limiting relative measure. Choosing the tolerances slowly enough gives Borel sets \(S_r\subset Q_r\) and numbers \(\zeta_r\to0\) such that \[ \frac{\mathop{\mathrm{vol}}_N(S_r)}{\mathop{\mathrm{vol}}_N(Q_r)}\ge e^{-o(r)},\qquad \sup_{n\in S_r}\|f_{r,x,n}-f\|_{Z_J}\le\zeta_r. \tag{57}\] To be explicit, choose tolerances \(1/j\) successively; postpone the \(j\)th choice until its probability lower bound has logarithm of absolute value at most \(r/j\). This also ensures that every other fixed compactness error tends to zero along the diagonal. The scheme and its compact were chosen before this diagonal.

We verify the support-volume hypothesis for recurrence. For an individual generator interval let \(\sigma_{j,r}\) denote the law of \(\exp(t e^{rs_i(E)}X_i)\). Since \(E\subset(1-c\delta)\operatorname{int}(\mathcal C)\), \[s_i(E)\le(1-c\delta)s_i(\mathcal C).\] Lemma 29 shows that \(k_0\mathop{\mathrm{supp}}(\sigma_{j,r})k_0^{-1}\) tends to the identity in normalized \(\mathcal C\) coordinates. The bounded polynomial product law therefore gives \[ \mathop{\mathrm{vol}}_N(Q_r\mathop{\mathrm{supp}}(\sigma_{j,r})) \le D_{j,r}\mathop{\mathrm{vol}}_N(Q_r),\qquad D_{j,r}=e^{o(r)}. \tag{58}\] All estimates hold with common envelopes for the fixed finite list.

For the \(j\)th switch, sample \(s_{j,1},s_{j,2}\) independently with law \(\sigma_{j,r}\), independently also for different \(j\), and put \(u_j=s_{j,1}s_{j,2}^{-1}\). Its law is \(\sigma_{j,r}*\check\sigma_{j,r}\). Apply Lemma 18 in the unimodular group \(N\) with the compact set \(Q_r\), the Borel set \(S_r\), and these increment laws. The mass bound (57), the denominator bound (58), and the fixed depth \(h\) give a starting point \(n_0\in S_r\) such that \[\mathbb P\{n_0u_1\cdots u_j\in S_r \text{ for every }1\le j\le h\}\ge e^{-o(r)}.\] Thus every intermediate sheet in a successful sequence has a good profile.

From parameter measure to endpoint volume. The successful interval parameters have Lebesgue measure at least \(e^{-o(r)}\) in the fixed product cube, since the uniform product density is a fixed constant. In \(E\)-normalized logarithmic coordinates, the endpoint map for the increments is a polynomial of fixed degree with convergent bounded coefficients and a submersive limit, by our choice of scheme. Lemma 17, or equivalently Corollary 19, now gives accessible endpoint volume in \(N\) at least \[ \exp\bigl(rP(E)-o(r)\bigr). \tag{59}\] If necessary, restrict the successful parameters to a compact subset of half their measure; this preserves the estimates and makes the image measurable. Left multiplication by \(n_0\) preserves the volume. The rank conclusion is a property of the full fixed polynomial map, established before the successful restriction.

Comparing source and image packing. Take \(d_p=\Delta_r z\) as pivot. For each prescribed generator increment, travel along \(\Delta_r z\exp_D(rt v)\) to a point where its weight attains \(s_i(E)\), perform the increment, and return to the pivot on the new sheet. Lemma 12 gives an \(o(r)\)-length switch at that point; the two \(D\) legs have length \(O(r)\). The difference of two interval parameters is a single generator increment. We therefore do not need a good profile at the algebraic midpoint of \(s_1s_2^{-1}\).

All sheet states at which a \(D\) leg is traversed belong to \(S_r\). The images of these legs consequently have heights in \[\pi F(x)+r\bigl(f(J)+o(1)\bigr).\] The short switches contribute only \(o(r)\) height errors by Lipschitzness. The finite concatenation has length \(O(r)\), begins at \(xn_0d_p\), and ends at the fixed \(D\) position \(d_p\) on one of the accessible sheets.

By Lemma 15, the endpoint volume (59) gives at least \(\exp(rP(E)-o(r))\) source points separated by any prescribed fixed large constant. Choose this constant so their quasi-isometry images remain separated. Their image paths have a common start, length \(O(r)\), and relative heights in \(r(f(J)-f(z)+o(1))\), since the starting profile value is \(f_{r,x,n_0}(z)=f(z)+o(1)\). The compact set \(f(J)-f(z)\) contains zero; enlarging it by a ball of radius tending to zero absorbs the error. Lemma 13, translation invariance of \(P\), and its continuity for Hausdorff distance give the upper bound \(\exp(rP(f(J))+o(r))\) for the same separated points. Taking logarithms and dividing by \(r\) proves (56). When \(\pi v=0\) the claimed lower bound is zero; it is also immediate from nonnegativity of \(P\). ◻

Proof. Horizontal derivatives take values in the finite group. Choose a basis of the first layer \(V\) and its left-invariant vector fields on \(D_\infty\). Flow-box Fubini and linewise Lipschitzness give locally bounded distributional derivatives of \(f\) in these directions. Their linear combinations define a measurable linear map \(M(z):V\to W\). For each fixed \(v\), it agrees almost everywhere with the line derivative in direction \(v\): both are the distributional derivative along that field. These assertions may be made simultaneously for a countable dense set of \(v\) by intersecting their full Haar-measure sets.

At a point of line differentiability, the image of a short segment, translated to start at zero and divided by its length, converges in Hausdorff distance to the linear segment of velocity \(M(z)v\). Indeed the definition of differentiability bounds the error uniformly on the entire sufficiently short one-sided interval. Differentiate (60) and use continuity and positive homogeneity of \(P\) to obtain \[ Mv\in\operatorname{conv}(\mathcal A\pi v),\qquad p(Mv)\ge p(\pi v) \tag{61}\] at the common full-measure points. Continuity in \(v\) extends these constraints from the dense set to all of \(V\).

The first constraint makes \(M\) vanish on \(\ker\pi\), so \(M=B\pi\) for a linear map \(B:W\to W\). It also gives \(p(Bw)\le p(w)\) by convexity, while the second constraint gives the reverse inequality. Thus \(B\) is a linear isometry of \(p\) and is invertible. The full linear isometry group of the polyhedral norm \(p\) is finite: its elements permute the finitely many vertices of the unit ball, and those vertices span \(W\). Average a Euclidean inner product over this full finite group. In its norm, \(Bw\) and every \(Aw\), \(A\in\mathcal A\), have the same length. Strict convexity and the convex-hull inclusion force \[Bw\in\{Aw:A\in\mathcal A\}\qquad(w\in W).\] If \(B\) differed from every \(A\in\mathcal A\), the finitely many proper linear subspaces \(\ker(B-A)\) could not cover \(W\). Hence \(B\in\mathcal A\). We have proved \[ M(z)=B(z)\pi,\qquad B(z)\in\mathcal A \quad\text{for almost every }z. \tag{62}\] Choose any label on the remaining null set, so \(B\) is a measurable finite-valued function throughout the domain.

Equality of support and variation fixes weight signs. Let \(v\) have generic height \(w=\pi v\), meaning \(\alpha_i(w)\ne0\) for all \(i\). Along almost every line of this direction, the absolutely continuous restriction \(h(t)=f(z\exp_\infty(tv))\) has derivative \(B(z\exp_\infty(tv))w\) almost everywhere. Thus \(p(h'(t))=p(w)\). We work on a compact interval \([0,T]\) inside such a line and translate \(h\) to have \(h(0)=0\), which does not change \(P\).

For a word basis \(\mathcal B\), let \(m_i(\mathcal B)\) count all occurrences of \(X_i\) in its words. Every generator is mandatory, so \(m_i(\mathcal B)\ge1\). Choose a basis maximizing \(P(h([0,T]))\), and put \(q_i(t)=(\alpha_i\circ h)'(t)\). Then \[\begin{align*} P(h([0,T])) &=\sum_i m_i(\mathcal B)\max_{0\le t\le T}\alpha_i(h(t)) \\ &\le\sum_i m_i(\mathcal B)\int_0^T(q_i(t))_+\,dt \le\int_0^T p(h'(t))\,dt =T p(w). \tag{63}\end{align*}\] The second inequality is the definition of \(p\) as the maximum over word bases. The segment volume inequality supplies the reverse inequality between the outer quantities, so every inequality in (63) is equality. Since every \(m_i(\mathcal B)\) is positive, each scalar maximum has no deficit. If \(t_i\) is a maximizing time for \(\alpha_i(h(t))\), this says \[ 0=\int_0^{t_i}(q_i(t))_-\,dt +\int_{t_i}^T(q_i(t))_+\,dt. \tag{64}\] Consequently a negative derivative cannot occur, on a set of positive measure, before a positive derivative. On this interval each weight’s positive-sign indicator is nonincreasing, up to null sets.

By Lemma 9, pullback by every label permutes the distinct oriented generator-weight rays. Genericity therefore makes every \(q_i\) nonzero almost everywhere, and the number of positive signs among those distinct rays is independent of the label \(B\). A finite collection of nonincreasing indicators with constant sum has each indicator constant almost everywhere. Thus all weight signs are constant along the line, on the interval under consideration. Applying this on a countable exhaustion of its open interval in \(\mathcal U_\delta\) shows the following: if \(w\) lies in a chamber of the generator-weight hyperplane arrangement, then almost all labels along the line send that chamber to the same chamber. Here each line meets \(\mathcal U_\delta\) in an interval because its projected height moves linearly through a convex set.

Extremal directions have constant horizontal derivative. The chambers just used are the connected components of \(W\setminus\bigcup_i\ker\alpha_i\). Their closures are pointed polyhedral cones, since the weights span \(W^*\). Let \(w\ne0\) span an extremal ray \(R\) of one of these closed cones, and choose a horizontal lift \(v\in V\).

For each chamber \(C_0\) whose closure contains \(R\), choose generic vectors \(v_j\to v\) with \(\pi v_j\in C_0\). We explain carefully why the almost-everywhere chamber agreement passes to this limiting direction. On a compact set of pairs \((z,s)\) whose flows stay in the domain, the maps \(z\mapsto z\exp_\infty(sv_j)\) preserve Haar measure and converge uniformly to the corresponding map for \(v\). Approximating the bounded measurable function \(B\) by continuous functions on a larger compact set shows that \[B(z\exp_\infty(sv_j)) \longrightarrow B(z\exp_\infty(sv)) \quad\text{in measure locally in }(z,s).\] The approximation errors are uniformly controlled by Haar invariance. Apply the same statement at a second time \(t\); Fubini gives convergence in measure locally in \((z,s,t)\). After a subsequence both labels converge almost everywhere. We restrict to compact sets of triples for which the segment between the two limiting positions stays in the open domain; for large \(j\) the nearby segments do so too. The preceding chamber argument applies to every \(v_j\). Since the set of labels is finite, the relation that two labels have the same image of \(C_0\) is closed. It follows that almost every such triple satisfies \[B(z\exp_\infty(sv))\overline C_0 =B(z\exp_\infty(tv))\overline C_0.\] There are only finitely many incident chambers. Intersect their full-measure sets, and then exhaust the domain of triples by compact sets.

The intersection of the closures of all chambers incident to \(R\) is exactly \(R\). Indeed the active hyperplanes of an extremal ray cut down to its line; the remaining chamber inequalities select the half-line containing \(w\). Equivalently, all hyperplanes through \(w\) allow both neighboring signs among the incident chambers, while the other hyperplanes have the fixed sign of \(w\). Intersecting the displayed equalities therefore shows that the two labels have the same image of \(R\). They preserve the invariant Euclidean norm chosen above, so their values on \(w\) are equal. Thus the derivative in direction \(v\) is constant at almost every pair of positions on almost every such line. Absolute continuity makes \(f\) affine on those lines.

When \(v\in\ker\pi\), the first inequality in (60) already makes \(f\) constant on every horizontal segment of direction \(v\).

Regularity makes the label constant on the domain. The extremal rays of a full-dimensional pointed chamber span \(W\). Choose finitely many lifts of such rays spanning a complement to \(\ker\pi\) in \(V\), and add a basis of \(\ker\pi\). The resulting vectors \(v_1,\ldots,v_b\) span \(V\). Let \(Y_j\) be their left-invariant horizontal fields on \(D_\infty\). The line affinities just proved imply \[Y_j^2 f=0\quad\text{in distributions},\qquad 1\le j\le b.\] For completeness, use flow coordinates for \(Y_j\) and test against a compactly supported smooth function. Along almost every flow line the second distributional derivative of an affine function is zero. Integration in the transverse coordinates gives the displayed identity. The flow preserves Haar measure, so no divergence term occurs.

The associated graded Lie algebra of \(D\) is generated by its first layer \(V\). Hence the fields \(Y_j\) and their iterated brackets span the full tangent space everywhere. Hörmander’s sum-of-squares theorem (Hörmander 1967, Theorem 1.1) applies to \(\sum_{j=1}^bY_j^2\), componentwise in \(W\), and makes \(f\) smooth. All hypotheses are local on the open domain: the fields are smooth and real, the bracket condition holds at every point, and the right side is the smooth function zero.

The horizontal derivative is now continuous. Its almost-everywhere membership in the finite set \(\{A\pi:A\in\mathcal A\}\) from (62) extends to every point, since a full-measure subset is dense. The domain is connected, so that derivative is one constant \(A\pi\). Every horizontal derivative of \(f-A\pi\) vanishes; the same is then true for iterated brackets. Bracket generation makes its entire differential zero, and connectedness gives \(f-A\pi=b\) for one constant \(b\in W\). ◻

Proof. Relative compactness and horizontal Lipschitz regularity were proved after Definition 28. Lemmas 31 and 32 supply both assumptions of Lemma 33 on every segment. That lemma gives (65). ◻

Proof. For a continuous function on \(Z\), its distance in the supremum norm to the set of restrictions of \(A\pi+b\) is a continuous, \(1\)-Lipschitz function of that continuous function. The affine restriction set is closed: take a subsequence with fixed \(A\), and evaluate at one point of \(Z\) to see that the offsets converge.

If (66) failed, there would be a sequence of maps, basepoints, and deterministic allowed suffixes for which the displayed probability is bounded below by a positive constant. Profile compactness supplies a subsequential law limit. Proposition 34 supports that limit on the affine class. The event whose probability is bounded below is closed, so the Portmanteau inequality is a contradiction.

For an independent random suffix, condition on its value. The bad probability is an average of deterministic bad probabilities, and hence is bounded by the same supremum, which tends to zero. For a compact exhaustion of \(\mathcal U_\delta\), the zero sets of the continuous affine-distance functions are closed. For each compact in the exhaustion and each \(\eta>0\), the Portmanteau inequality for the open event that this distance exceeds \(\eta\) shows that its limiting probability is zero. Thus every limiting law is supported on functions affine on all these compact sets. Use a constant-label subsequence and evaluate at a fixed point to see that their labels and offsets are compatible across the exhaustion; equivalently, the class of functions affine on the entire domain is closed in \(C_{\mathrm{loc}}(\mathcal U_\delta,W)\).

The function \(\varepsilon(r)\) was fixed before taking the limit. This assertion is not a supremum over all possible subexponential envelopes at one finite scale. Nor does it assert concentration near one particular pair \((A,b)\): different affine profiles may still occur in the limiting law. ◻

Proof. Bracketing a mixed word with any generator produces another mixed word. The derivation identity and the Jacobi identity therefore show that their span is the ideal generated by the outside letters. Homogeneous bracket words span \(\mathfrak n\). If a weight \(\gamma\) lies outside \(S\), a word of weight \(\gamma\) cannot be pure: a sum of inside weights belongs to \(S\). Consequently the entire generalized weight space \(\mathfrak n_\gamma\) lies in \(\mathfrak i_S\) whenever \(\gamma\notin S\). For \(v\in\mathfrak d\) and an outside letter \(X_i\), the vector \([v,X_i]\) stays in \(\mathfrak n_{\alpha_i}\) and hence in \(\mathfrak i_S\). Applying the derivation identity to words proves \([\mathfrak d,\mathfrak i_S]\subset\mathfrak i_S\). This argument includes the nilpotent parts of the generalized weight actions.

Both possible cones \(S\) are pointed. A nonempty sum of their nonzero inside weights cannot be zero. The quotient algebra is spanned by pure words, so it has no zero generalized weight space. The zero weight space of \(\mathfrak n\) is therefore contained in \(\mathfrak i_S\); in particular \(\mathfrak d\cap\mathfrak n\subset\mathfrak i_S\). The exponential descriptions of the subgroups give \(N\cap D\subset I_S\). The ideal exponentiates to a closed subgroup because the exponential map is a diffeomorphism and \(\mathfrak i_S\) is a closed linear subspace. The Baker–Campbell–Hausdorff group on \(\mathfrak n/\mathfrak i_S\) identifies with \(N/I_S\): the algebra quotient induces a surjective group homomorphism with kernel \(I_S\). This realizes \(L_S\) as a connected, simply connected nilpotent Lie group. A linear section of the algebra quotient, composed with its logarithm and the exponential map of \(N\), gives a continuous section \(L_S\to N\). Images of compact sets under this section are compact, proving the lifting assertion.

The action of \(N\rtimes D\) on \(L_S\) is \[(n,d)\cdot(xI_S)=n(dxd^{-1})I_S.\] The kernel of the multiplication map \(N\rtimes D\to G\) consists of \((h^{-1},h)\) with \(h\in N\cap D\). Its action sends \(xI_S\) to \(xh^{-1}I_S=xI_S\). Thus the action factors through \(G\), without requiring that the multiplication map be injective. If \(g=nd\), its value at the identity is \(nI_S\), proving the stabilizer assertion and the stated equivariance of \(p_S\).

Translations by \(N\) preserve quotient Haar measure. The determinant of the \(D\)-action is the exponential of its trace character; its unipotent parts have determinant one. The weight description thus gives (71). To check its positivity assertion, project to the abelianization of \(\mathfrak n\). The image of \(\mathfrak i_S\) there is exactly the span of the outside generator images. The inside generator images remain linearly independent in the quotient abelianization. They can be included in a basis of \(\mathfrak n/\mathfrak i_S\) selected from pure words. Summing that basis’s weights proves the asserted integral combination. Every inside weight in a two-generator cone is a nonnegative real combination of its two defining weights, and each defining generator itself survives. Both extremal coefficients are therefore positive. ◻

Proof. If \(Z\) is mixed, adjoining occurrences of \(X_i\) does not change its width on \(T_*\), because \(s_i(T_*)=0\). In \[\mathop{\mathrm{Ad}}_{k_q}Z =\sum_{p=0}^{m-1}\frac{t_q^p}{p!}(\mathop{\mathrm{ad}}X_i)^p Z,\] every nonzero term is mixed and has the same \(c_Z\). Its added exponential cost is at most \(pM\delta r+o(r)\). Expanding a word in a maximal-width ideal basis uses only basis vectors whose widths are at least its own, by basis exchange. Consequently the positive differences \[(\epsilon_1-\epsilon_2)c_Z, \qquad (\epsilon_+-\epsilon_1)c_Z\] absorb all those costs if \(\delta\) is sufficiently small. There are only finitely many words and \(p<m\). Their positive gaps also absorb the common \(o(r)\) envelope. We obtain \[k_qB^I_r(T_2,1)k_q^{-1}\subset B^I_r(T_1,O(1)), \qquad k_q^{-1}B^I_r(T_1,1)k_q\subset B^I_r(T_+,O(1)).\]

Put \(Q_r=k_qB^I_r(T_2,1)k_q^{-1}\). Choose a maximal disjoint family \(i_\kappa Q_r\) with anchors in \(B^I_r(T_1)\). Such a family is finite: each set lies in \(B^I_r(T_1,O(1))\), by the first inclusion and normalized multiplication. Conjugation by \(k_q\) preserves Haar on \(I_S\), since its derivative is the exponential of the nilpotent map \(\mathop{\mathrm{ad}}(t_qX_i)|_{\mathfrak i_S}\) and has determinant one. Dividing volumes in (73) bounds the number of anchors by \(O(e^{r(P_1-P_2)})\). Maximality implies that every translate \(iQ_r\), for \(i\in B^I_r(T_1)\), meets some \(i_\kappa Q_r\). Hence \[B^I_r(T_1)\subset\bigcup_\kappa i_\kappa Q_rQ_r^{-1} \subset\bigcup_\kappa i_\kappa k_qB^I_r(T_2,O(1))k_q^{-1}.\] Assign a point to the first set containing it. This gives the measurable partition, and the second inclusion above proves (77).

For the last assertion write \(u=i\exp_N\sigma(s)\), where \(i\in B^I_r(T_2,e^{o(r)})\) and \(\|s\|\le e^{o(r)}\), using (75). Factor \[\exp_N\sigma(s)e_r=i_{e_r}\exp_N\sigma(s_{e_r}).\] The polynomial coordinate maps show that \(\|\log i_{e_r}\|+\|s_{e_r}\|\le e^{M'\delta r+o(r)}\) for a fixed multiple \(M'\) of \(M\). Every ideal basis width on \(T_2\) is strictly positive. Decrease \(\delta\) so that \(M'\delta\) is less than all these widths. Then \(i_{e_r}\in B^I_r(T_2,O(1))\) for all sufficiently large \(r\), and the ideal coordinate \(ii_{e_r}\) has the required bound. Finitely many simultaneous requirements have a positive common threshold for \(\delta\). ◻

Proof. Take \(\delta\) below the threshold in Proposition 36. The two profiles in a limiting support then have the form \[h_1(z)=A\pi z+b_1,\qquad h_2(z)=A\pi z+b_2 \quad(z\in\mathcal U_\delta)\] with one \(A\in\mathcal A\). Fix such a pair and a subsequence whose joint laws converge to a law containing it in its support. It suffices to prove \(b_1=b_2\). There are finitely many letters and labels, so these may be fixed after passing to a subsequence. Write \(\alpha=\alpha_i\).

Proof. Since \(\mathcal A^*\) permutes the generator rays, the source quotient is defined. Choose \(v\in\mathfrak d\) with \(\alpha(\pi v)<0\). There are only finitely many ray and label choices, so choose all these \(v\)’s once.

Convergence and a uniform compact shadow. For \(x_t=x\exp_D(tv)\), Proposition 44 gives \[\beta\bigl(\pi F(x_t)-\pi F(x)\bigr) =t\alpha(\pi v)+o(t)\le-c_0t \qquad(t\ge T_0),\] with \(c_0,T_0>0\) uniform in \(F,x\) and the finite choices. Set \(y_t=F(x)^{-1}F(x_t)\). Its distance from \(1\) is \(O(1+t)\). By Lemma 13 it has a lift \(y_t=n_td_t\) with \(|d_t|_D\le C(1+t)\). The increments \(y_t^{-1}y_{t+1}\) have uniformly bounded length and hence bounded lifts. Equations (95) and (96) imply \[\left\|\log\bigl(p_\beta(y_t)^{-1}p_\beta(y_{t+1})\bigr)\right\| \le C(1+t)^M e^{-c_1t}.\] The series is summable. In a left invariant Riemannian metric, \(d(1,\exp Z)\le C\|Z\|\), by using the path \(s\mapsto\exp(sZ)\). Thus the projected trajectory at integer times is Cauchy and stays in a fixed closed ball. Completeness and properness of the metric give convergence in a fixed compact set. The same estimate for time increments of size at most one gives convergence for real \(t\to\infty\). The initial interval \(0\le t\le T_0\) contributes a fixed compact set, so the limit lies in a fixed compact \(Q_\beta\). Equivariance of \(p_\beta\) under the \(G\)-action therefore gives a limit \[ b(x)=\lim_{t\to\infty}p_\beta F(x\exp_D(tv)) \in F(x)\cdot Q_\beta . \tag{98}\]

Independence of the representative. We first record the path estimate used for this purpose. Suppose, for large \(t\), a source path stays at distance at most \(C_2t\) from a fixed \(x\), has length bounded by a polynomial in \(t\), and satisfies \[\alpha\bigl(\pi(x^{-1}z)\bigr)\le-c_2t\] at each of its points \(z\). The uniform sublinear estimate puts its image, relative to \(F(x)\), at \(\beta\)-heights at most \(-c_2t/2\) for all sufficiently large \(t\). The lifts at all image points have \(D\)-length \(O(t)\), because their distance from \(F(x)\) is \(O(t)\). Subdivide the source path into bounded-length pieces. Equations (95) and (96) bound each projected image increment by a polynomial times \(e^{-ct}\). There are only polynomially many pieces, so the projected distance between the endpoints tends to zero. Constants here may depend on this fixed collection of paths.

For fixed \(d\in D\), connect \(x\exp(tv)\) to \(xd\exp(tv)\) by \(xd(s)\exp(tv)\), where \(d(s)\) is a fixed piecewise smooth path in \(D\) from \(1\) to \(d\). Its length is polynomial in \(t\), since conjugation in the nilpotent group \(D\) has polynomial norm. Its points have distance \(O(t)\) from \(x\), and their relative \(\alpha\)-heights equal \(t\alpha(\pi v)+O(1)\). The preceding path estimate gives \(b(xd)=b(x)\).

For fixed \(u\in I_\alpha\), we will vary the height within the halfspace \(\alpha<0\) to realize \(u\) while keeping the \(\alpha\)-height strongly negative. Choose a compact convex set \(E\subset W\) containing \(\pi v\), entirely in \(\alpha<0\), such that every bracket word containing an outside-ray letter has \(h_Z(E)>0\). Here is why such an \(E\) exists. Every generator form outside the positive \(\alpha\)-ray is unbounded above on the halfspace \(\{\alpha\le\alpha(\pi v)\}\). For each outside form choose a point in that halfspace on which it is as large as desired, and take the convex hull together with \(\pi v\). The support maxima for inside-ray letters remain fixed negative numbers. Since the list of words and their lengths is finite, sufficiently large outside maxima make the support sum of every mixed word positive.

By Lemma 38, the mixed words span \(\mathop{\mathrm{Lie}}(I_\alpha)\). Thus the fixed element \(u\) belongs to \(B_t(E,O(1))\) for all sufficiently large \(t\). One can see this directly in the all-word description of the box: the fixed coordinates of \(\log u\) in a mixed-word basis are dominated by its positive exponential widths. Choose an affine section \(\sigma:W\to V\) of \(\pi\) with \(\sigma(\pi v)\) equal to the abelianization of \(v\). The compact convex set \(E_V=\sigma(E)\) contains that abelianization and satisfies \(\pi E_V=E\). The paths of Lemma 12, applied to \(E_V\) and the pivot \(\exp(tv)\), connect \(\exp(tv)\) to \(u\exp(tv)\) in length \(O(t)\) with heights in \(t(E+o(1))\). All their heights remain strictly negative in \(\alpha\) by a linear amount, and all their points have distance \(O(t)\) from \(1\). Left translation by \(x\) and the preceding path estimate give \(b(xu)=b(x)\).

The stabilizer of \(1\in L_\alpha\) is \(I_\alpha D\). The two invariances therefore show that \(b(x)\) depends only on \(p_\alpha x\). Define \(f_\beta(p_\alpha x)=b(x)\). This establishes (97).

Inverse and continuity. By Proposition 44, \(A_{\bar F}=A^{-1}\), so the construction for \(\bar F\) gives \(\bar f:L_\beta\to L_\alpha\). For \(x_t=x\exp(tv)\) the quotient point \(p_\alpha x_t\) is fixed. Equation (97) at \(x_t\) supplies representatives \(z_t\) of \(f_\beta(p_\alpha x)\) within a fixed distance of \(F(x_t)\): lift the compact set \(Q_\beta\) to a bounded set in \(G\). The coarse-inverse estimate and the shadow estimate for \(\bar f\) then give \[\bar f(f_\beta(p_\alpha x))\in x_t\cdot Q'\] for a fixed compact \(Q'\subset L_\alpha\). The sets on the right contract to \(p_\alpha x\), since \(\alpha(\pi v)<0\). Thus \(\bar f f_\beta=\mathop{\mathrm{id}}\). Interchanging the maps proves \(f_\beta\bar f=\mathop{\mathrm{id}}\).

Fix \(x\) and \(t\). The projection of a fixed-radius neighborhood of \(x_t\) contains a neighborhood of \(p_\alpha x\); this uses only that the quotient projection is a submersion. By the Lipschitz bound and (97), its image under \(f_\beta\) is contained in \(F(x_t)\cdot Q''\), with \(Q''\) fixed. These sets have diameter tending to zero. Indeed, after normalization by \(F(x)\), the \(D\)-part has length \(O(t)\) and negative linear \(\beta\)-height, so (96) applies; quotient left translations are isometries. The fixed action of \(F(x)\) is Lipschitz, being a left translation followed by an automorphism. This proves continuity at \(p_\alpha x\). The same argument proves continuity of \(\bar f\), completing the proof. ◻

Proof. Fix a unit ball \(V_-\subset L_\beta\) and a bounded set of lifts of radius \(R_-\) in \(G\). Denote the common Lipschitz constants by \(L_F,L_{\bar F}\) and the common coarse-inverse error by \(E_0\). For each such lift \(b\), \[d_G(\bar F(F(x)b),x)\le L_{\bar F}R_-+E_0.\] The shadow estimate for the inverse boundary map therefore puts \(f_\beta^{-1}(F(x)\cdot V_-)\) inside \(x\cdot U\), where \(U\) is one compact ball containing \[\{h\cdot q:|h|_G\le L_{\bar F}R_-+E_0,\ q\in Q_{\rm inv}\}.\] Here \(Q_{\rm inv}\) is its fixed shadow compact. This choice of \(U\) is independent of \(x\).

Choose a bounded lift set for this \(U\) once, of radius \(R_U\). For every lift \(b\) in it, \(d_G(F(xb),F(x))\le L_FR_U\). The forward shadow estimate gives \(f_\beta(x\cdot U)\subset F(x)\cdot V_+\) for one compact \(V_+\). Thus \[F(x)\cdot V_-\subset f_\beta(x\cdot U) \subset F(x)\cdot V_+.\] Both comparison sets have positive finite Haar volume. The \(G\)-action scales that volume by \(e^{\ell_\beta(\pi F(x))}\), proving the assertion. There are finitely many ray and label choices, so all constants can be chosen uniformly. ◻

Proof. Fix a generator \(X_i\) and any \(t\in\mathbb R\). For every source generator ray other than that of \(\alpha_i\), the element \(\exp(tX_i)\) lies in its ideal and acts trivially on the corresponding quotient. Hence the two source sets \[x\exp(tX_i)\cdot U,\qquad x\cdot U\] are identical. Lemma 46 bounds the difference of the associated target trace heights by a uniform constant. By (94), this bounds every target ray-height difference except possibly the ray paired with \(\alpha_i\).

There is a relation \(\sum_\rho c_\rho\rho=0\) over all distinct target generator rays, with every \(c_\rho>0\), by Lemma 9. Applying it to the height difference bounds the one remaining form as well. Since the ray forms span \(W^*\), there is a uniform constant \(B_1\) with \[|\pi F(x\exp(tX_i))-\pi F(x)|\le B_1\] for every \(x,i,t\). No bound on \(t\) is imposed.

The sets \(B_r(\mathcal C)\) exhaust \(N\), since every generator support maximum on \(\mathcal C\) is positive. By Lemma 10, every such box element is a product of at most one fixed number \(M\) of generator exponentials. Their parameters can be arbitrary, because the preceding estimate is parameter independent. Thus \(B_N=MB_1\) works for every \(n\). Finally \(nd=d(d^{-1}nd)\) and \(d^{-1}nd\in N\); applying the bound at the basepoint \(d\) proves the last assertion without introducing any dependence on \(d\). ◻

Proof. For compact \(E\subset B(R_E)\) and \(T>R_E\), Tonelli’s theorem gives \[\int_{B(T)}\rho(zE)\,d\mu(z) =\int_L\mu(B(T)\cap yE^{-1})\,d\rho(y).\] Nilpotent groups are unimodular, so inversion preserves Haar measure. The inner quantity equals \(\mu(E)\) for \(y\in B(T-R_E)\), vanishes outside \(B(T+R_E)\), and lies between zero and \(\mu(E)\) everywhere. Consequently \[ \mu(E)\rho(B(T-R_E)) \le \int_{B(T)}\rho(zE)\,d\mu(z) \le \mu(E)\rho(B(T+R_E)). \tag{100}\]

Fix a large-scale doubling constant \(D_0\) such that \(\mu(B(2T))\le D_0\mu(B(T))\) for all sufficiently large \(T\). Such a constant follows from the polynomial volume estimates for nilpotent groups, as in Lemma 6. Put \(q_0=a_0/\mu(E_0)\). Apply (100) to \(E_0\), first with averaging radius \(T-R_{E_0}\) and then with radius \(T+R_{E_0}\). The assumed translate bounds and doubling give, for sufficiently large \(T\), \[(KD_0)^{-1}q_0\mu(B(T)) \le\rho(B(T))\le KD_0q_0\mu(B(T)).\] For a fixed \(j\), choose \(T\) so large that these bounds apply to \(T-R_{E_j}\) and \(T+R_{E_j}\) and that \(T\ge2R_{E_j}\). Use (100) for \(E_j\) and the assumed translate bounds once more. Doubling yields \[(K^2D_0^2)^{-1}q_0 \le\frac{a_j}{\mu(E_j)} \le K^2D_0^2q_0.\] The required averaging radius may depend on \(j\); the displayed constants do not. Only integration of nonnegative measurable functions was used, so the proof applies equally to singular Radon measures. ◻

Proof of Theorem 4. If \(W=0\), the assertion is immediate. Otherwise first take a Lipschitz pair from the uniform family of Lemma 21, with label \(A_F\) from Proposition 44. Fix a target ray \(\beta\) and its source ray \(\alpha=\beta\circ A_F\). Let \(f_\beta\) and \(U\) be given by Lemmas 45 and 46. Define a Radon measure on \(L_\alpha\) by \[\rho(E)=\mathop{\mathrm{vol}}_{L_\beta}(f_\beta(E)).\] This is a measure because \(f_\beta\) is a homeomorphism; no absolute-continuity property is asserted.

For \(d\in D\), put \(U_d=d\cdot U\) and \(a_d=e^{\ell_\beta(\pi F(d))}\). Every \(z\in L_\alpha\) has a representative \(n\in N\), and \(zU_d=nd\cdot U\). Lemmas 46 and 47 give \[\rho(zU_d)\asymp a_d \qquad(z\in L_\alpha,d\in D)\] with one uniform comparison constant. Lemma 48, applied to these sets and the reference set \(U_1=U\), implies \[\frac{e^{\ell_\beta(\pi F(d))}}{\mathop{\mathrm{vol}}_{L_\alpha}(U_d)} \asymp \frac{e^{\ell_\beta(\pi F(1))}}{\mathop{\mathrm{vol}}_{L_\alpha}(U)}.\] The source action scales Haar by \(e^{\ell_\alpha(\pi d)}\). Taking logarithms therefore gives \[ \ell_\beta(\pi F(d)-\pi F(1)) =\ell_\alpha(\pi d)+O(1) \qquad(d\in D), \tag{101}\] with a uniform error.

We identify its linear term using the already proved sublinear estimate. For \(w\in W\), choose \(H\in\mathfrak d\) with \(\pi H=w\) and set \(d_t=\exp_D(tH)\). Its length is \(O(t)\), so (101) and Proposition 44 give \[t\ell_\alpha(w)+O(1) =\ell_\beta(\pi F(d_t)-\pi F(1)) =t\ell_\beta(A_Fw)+o(t).\] After division by \(t\), \(\ell_\alpha=\ell_\beta\circ A_F\). The target ray traces span \(W^*\) by (94). A fixed spanning subfamily of (101) therefore yields \[|\pi F(d)-\pi F(1)-A_F\pi d|\le B_D \qquad(d\in D),\] with uniform \(B_D\). For any \(g=nd\), Lemma 47 and \(\pi n=0\) extend this bound to \(g\), with bound \(B_D+B_N\). All finite ray and label choices were absorbed into the constants, so the conclusion holds for the whole fixed family.

Finally an original \((K,C)\) quasi-isometry is at a uniformly bounded distance from its Lipschitz approximation by Lemma 21. Since \(\pi\) is Lipschitz, changing back alters the error by a uniform constant, proving the stated theorem for the original maps. ◻

Proof. Dekimpe’s lattice realization theorem (Dekimpe 2000, Theorem 4.1) gives \(\Gamma\) and \(S\), with \(\Gamma\) normal of finite index in \(P\) if desired. The uniform lattice acts properly and cocompactly by left translations on \(S\). Thus its inclusion is a quasi-isometry by the Milnor–Švarc lemma; the same observation applies to a finite-index subgroup acting on a Cayley graph. See (Cornulier and Harpe 2016, Theorem 4.C.5(1) and Proposition 4.C.11(3)). Cornulier–Tessera’s reduction (Cornulier and Tessera 2017, Lemma 3.A.1) gives a real-triangulable \(G\) quasi-isometric to \(S\). It remains to check unimodularity and the stated properties of \(H\) and \(\Gamma\).

The group \(P\) is amenable, since cyclic groups are amenable and this property is preserved by extensions and finite extensions. We give the metric Følner argument that transfers this information to \(G\) and \(H\). It is important here to obtain metric Følner sets in \(G\), rather than only amenability as a locally compact group; compare (Tessera 2008, Proposition 11.11 and Corollary 11.13).

Let \(Y\) denote either \(G\), with a left invariant Riemannian metric and left Haar measure \(\mu\), or \(H\), with a word metric and counting measure. Choose a quasi-isometry \(u:P\to Y\) with coarse-surjectivity radius \(c>0\), and fix \(a>c\). For finite nonempty \(E\subset P\), put \[A_E=\bigcup_{p\in E} B_Y(u(p),a).\] For each fixed radius, the lower quasi-isometry inequality bounds the number of points of \(P\) whose images lie in a ball of that radius. Consequently the balls in this union have bounded multiplicity, and \[ \mu(A_E)\geq c_1|E| \tag{102}\] for a constant \(c_1>0\) independent of \(E\).

Write \(N_R(A)\) for the metric \(R\)-neighborhood of a set \(A\). If \(y\in N_R(A_E)\setminus A_E\), choose \(p_y\in P\) with \(d_Y(y,u(p_y))\leq c\) and \(p\in E\) with \(d_Y(y,u(p))\leq R+a+1\). Since \(a>c\), we have \(p_y\notin E\). The lower quasi-isometry inequality gives \(d_P(p_y,E)\leq R'\), with \(R'\) depending only on \(R,a\) and the quasi-isometry constants. It follows that \[ \mu\bigl(N_R(A_E)\setminus A_E\bigr) \leq \mu(B_Y(1,c))\,|N_{R'}(E)\setminus E|. \tag{103}\] An increasing-radius Følner sequence in \(P\), together with (102)–(103), therefore gives finite-measure sets \(A_j\subset Y\) satisfying \[ \frac{\mu(N_R(A_j))}{\mu(A_j)}\longrightarrow1 \qquad\text{for every fixed }R. \tag{104}\] For the discrete group \(H\), this is the Følner criterion for amenability.

For \(Y=G\), right translation by \(g\) moves every point a distance \(d_G(1,g)\), so \(A_jg\subset N_{d_G(1,g)}(A_j)\). If right translation multiplied left Haar measure by a factor greater than one, this inclusion would contradict (104). A nontrivial modular homomorphism has such a value, after replacing \(g\) by \(g^{-1}\) if necessary. Thus \(G\) is unimodular.

Finally, a connected simply connected solvable Lie group of dimension \(n\) is diffeomorphic to \(\mathbb R^n\); explicit coordinates are given in (Cornulier and Tessera 2017, Lemma 3.B.2). The compact smooth manifold \(\Gamma\backslash S\) is therefore aspherical and has a finite CW structure. It is a finite classifying space for \(\Gamma\), and manifold Poincaré duality makes \(\Gamma\) an integral Poincaré duality group of dimension \(n\). ◻

Proof. Fix a finitely generated subgroup \(J\leq K_0\) and a finite symmetric generating set \(T\) of \(J\). The heights of all orbit points \(f(j)\), \(j\in J\), have norm at most \(D\). For a word \(j=t_1\cdots t_m\) with \(m\leq t\), put \(j_i=t_1\cdots t_i\). The upper quasi-isometry inequality gives a constant \(L_T\) such that \[d_G(f(j_{i-1}),f(j_i))\leq L_T \qquad(1\leq i\leq m).\] Join these successive points by minimizing geodesics. The resulting path starts at \(1\), ends at \(f(j)\), and has length at most \(L_Tt\). The homomorphism \(\pi\) is Lipschitz for a left invariant Riemannian metric: its differential has constant norm in left-trivialized tangent spaces. If \(L_\pi\) is a Lipschitz constant, every point of the path has height norm at most \(D+L_\pi L_T\).

Lemma 14, with these fixed length and height bounds, shows that a maximal \(1\)-separated subset of \(f(B_T(t))\) has at most \(C_T(1+t)^{b_T}\) points. The unit balls about that subset cover \(f(B_T(t))\). The lower quasi-isometry inequality bounds the number of elements of \(H\) mapping into each such ball by one constant: any two such elements are at uniformly bounded distance in the fixed word metric of \(H\). Hence \[|B_T(t)|\leq C'_T(1+t)^{b_T}.\] This is an intrinsic growth bound for \(J\); no undistortion of \(J\) in \(H\) is assumed. Gromov’s polynomial-growth theorem (Gromov 1981, Main Theorem, p. 54) makes \(J\) virtually nilpotent. ◻

Proof. Choose quasi-inverse maps \(f:H\to G\) and \(r:G\to H\). Bounded changes allow us to arrange \(f(e)=1\) and \(r(1)=e\). For \(h\in H\), define \[F_h=f\circ L_h\circ r, \qquad L_h(k)=hk.\] The maps \(F_h\) have uniform quasi-isometry constants, because the maps \(L_h\) are isometries. There is a constant \(D_0\) such that \[ \sup_{g\in G}d_G(F_hF_k(g),F_{hk}(g))\leq D_0 \qquad(h,k\in H), \tag{106}\] and \(F_e\) is uniformly close to the identity. These assertions follow from the bounded errors in \(r\circ f\) and \(f\circ r\). Also \(F_h(1)=f(h)\).

Put \(s(h)=\pi f(h)\). The assumed height theorem gives a single \(B<\infty\) and matrices \(A_h\) in the fixed finite group \(\mathcal A_G\) such that \[ \|\pi F_h(g)-s(h)-A_h\pi(g)\|\leq B \qquad(h\in H,\ g\in G). \tag{107}\] Apply this formula twice to \(F_hF_k(g)\), and compare with its application to \(F_{hk}(g)\) using (106). It follows that \[(A_hA_k-A_{hk})\pi(g)\] is bounded as \(g\) ranges over \(G\). Since \(\pi\) is surjective onto the vector space \(W\), its linear coefficient vanishes. Similarly \(A_e=I\). Thus \(h\mapsto A_h\) is a homomorphism, and \[H_0=\{h\in H:A_h=I\}\] is normal and has finite index. Evaluating the same comparison at \(g=1\) gives a constant \(D_1\) such that \[ \|s(hk)-s(h)-s(k)\|\leq D_1\qquad(h,k\in H_0). \tag{108}\] The uniform constant in the height theorem is essential here: \(D_1\) does not depend on \(h\) or \(k\).

The subgroup \(H_0\) is amenable. Let \(m\) be a left-invariant mean on bounded real functions on \(H_0\), extended coordinatewise to functions with values in \(W\). Define \[ \tau(h)=m_x\bigl(s(hx)-s(x)\bigr),\qquad h\in H_0. \tag{109}\] For fixed \(h\), the function being averaged is bounded, since (108) bounds its distance from \(s(h)\) by \(D_1\). Positivity of the mean gives \[ \|\tau(h)-s(h)\|\leq D_1. \tag{110}\] To verify additivity, write \[s(hkx)-s(x)=s(hkx)-s(kx)+s(kx)-s(x).\] Left invariance under \(x\mapsto kx\) shows that the means of the two terms on the right are \(\tau(h)\) and \(\tau(k)\). Therefore \(\tau(hk)=\tau(h)+\tau(k)\).

Let \(K_0=\ker\tau\). Equation (110) is the hypothesis of Lemma 50; that lemma makes each finitely generated subgroup of \(K_0\) virtually nilpotent. Nilpotent groups are elementary amenable by their central series, and so are their finite extensions. The finitely generated subgroups of \(K_0\) form a directed family with union \(K_0\). Thus \(K_0\) is elementary amenable. The exact sequence \[1\longrightarrow K_0\longrightarrow H_0 \overset{\tau}{\longrightarrow}\tau(H_0)\longrightarrow1\] and abelianness of \(\tau(H_0)\) show that \(H_0\) is elementary amenable. Finally \(H/H_0\) is finite, so \(H\) is elementary amenable as well. ◻

Proof of Theorem 1. Let \(P\) and \(H\) satisfy the hypotheses of the theorem. Apply Lemma 49 and retain both the real-triangulable unimodular model \(G\) and the uniform lattice \(\Gamma\leq S\).

If \(W=0\), Lemma 7 makes \(G\) nilpotent. Its metric balls have polynomial Haar volume, by the nilpotent coordinate bounds of Lemma 6. A quasi-isometry from \(H\) to \(G\) has bounded multiplicity on bounded scales, so disjoint small-ball packing transfers this to polynomial word growth of \(H\). Gromov’s theorem (Gromov 1981, Main Theorem, p. 54) makes \(H\) virtually nilpotent, hence virtually polycyclic. This also includes the zero-dimensional model: in that case \(H\) is finite. We may therefore suppose \(W\ne0\).

By Lemma 49, \(H\) is amenable. Apply Theorem 4 to the uniform family transferred from its left translations. Proposition 51 shows that \(H\) is elementary amenable.

The group \(\Gamma\) has a finite classifying space and is quasi-isometric to \(H\). Li’s coarse invariance theorem (Li 2018, Corollary 4.47), applied over \(\mathbb Z\) for each \(n\), gives \(H\in\mathrm{FP}_\infty(\mathbb Z)\). The elementary amenable finiteness theorem (Kropholler et al. 2009, Theorem 1.1, (viii)\(\Rightarrow\)(v)) now gives a finite-index subgroup \(V\leq H\) of type \(F\). The structural conclusion recorded in Section 2 of the same paper also gives a finite-index solvable subgroup \(U\leq H\). Set \(H_1=U\cap V\). Then \(H_1\) is solvable and has finite index in \(H\). The finite cover of a finite classifying space for \(V\) corresponding to \(H_1\) is a finite classifying space for \(H_1\). In particular, \[\operatorname{cd}_{\mathbb Z}H_1<\infty.\] This also implies that \(H_1\) is torsion-free: restriction of a finite projective resolution to a hypothetical nontrivial finite cyclic subgroup would give finite integral cohomological dimension for that subgroup, which is impossible.

Both \(H_1\) and \(\Gamma\) now have finite integral cohomological dimension, and they are quasi-isometric. Since \(\Gamma\) is an integral Poincaré duality group, Li’s duality transfer (Li 2018, Corollary 4.48), with coefficient ring \(\mathbb Z\), makes \(H_1\) an integral Poincaré duality group. Here the type-\(F\) subgroup supplies the finite-cohomological-dimension hypothesis. Bieri’s theorem (Bieri 1981, Theorem 9.23) says that a solvable integral Poincaré duality group is polycyclic. Hence \(H_1\) is polycyclic, and \(H\) is virtually polycyclic. ◻

Proof of Corollary 2. Let \(\Lambda\) be a lattice in a connected simply connected solvable Lie group \(S\). Such a lattice is polycyclic (Farb and Weinberger 2008, Proposition 3.4), and is uniform (Geng 2015, Theorem 2.9(i)). It is therefore finitely generated and quasi-isometric to \(S\) by the Milnor–Švarc lemma. A finitely generated group \(H\) quasi-isometric to \(\Lambda\), or equivalently to \(S\), is virtually polycyclic by Theorem 1. Dekimpe’s realization theorem (Dekimpe 2000, Theorem 4.1) then embeds a finite-index subgroup of \(H\) as a uniform lattice in some connected simply connected solvable Lie group.

Conversely, every virtually polycyclic comparison group has such a finite-index lattice model by the same realization theorem. Finite index preserves quasi-isometry, so the lattice formulation implies the group formulation. The ambient Lie group obtained for \(H\) is not prescribed by this equivalence. ◻