Trace rank and the idempotent conjecture

For a torsion-free group the identity is the only finite-order element, so Theorem 1 also has numerical and ring-theoretic consequences. Define the canonical trace and the augmentation by \[\tau_G\!\left(\sum_{g\in G}a_g g\right)=a_1,\qquad \epsilon_G\!\left(\sum_{g\in G}a_g g\right)=\sum_{g\in G}a_g.\] For a matrix \(b=(b_{ij})\in M_q(\mathbb CG)\), put \(\tau_{G,q}(b)=\sum_{i=1}^q\tau_G(b_{ii})\), and let \(\epsilon_{G,q}(b)\in M_q(\mathbb C)\) be the entrywise augmentation. The coefficient at \([1]_G\) of the Hattori–Stallings trace defines \[\tau_{G,*}:K_0(\mathbb CG)\longrightarrow\mathbb C,\qquad \tau_{G,*}([e(\mathbb CG)^q])=\tau_{G,q}(e).\] The augmentation also defines the integer-valued augmentation rank \[\operatorname{rk}_{\epsilon}:K_0(\mathbb CG) \xrightarrow{(\epsilon_G)_*}K_0(\mathbb C) \xrightarrow{\dim_{\mathbb C}}\mathbb Z.\]

The idempotent conjecture commonly attributed to Kaplansky asks whether, for every torsion-free group \(G\) and every field \(k\), an element \(e\in kG\) satisfying \(e^2=e\) must be \(0\) or \(1\); see (Öinert and Wagner 2023, Problem 3). It concerns elements of the group ring itself, not arbitrary idempotent matrices over that ring.

Part (ii) resolves the characteristic-zero Kaplansky idempotent conjecture positively, and includes coefficient domains as well as fields. Section 10 derives both parts from the trace theorem. The integer range in part (i) concerns algebraic \(K_0(\mathbb CG)\); its relation to the reduced \(C^*\)-completion is made explicit there.

The integral trace consequence also links the theorem to fixed-point theory. A map \(f\) is a homotopy idempotent if \(f\circ f\simeq f\). Berrick, Chatterji and Mislin’s equivalences (Berrick et al. 2007, Theorems 1–2) then imply that every homotopy idempotent on a connected closed smooth oriented manifold of dimension greater than two is freely homotopic to a map with exactly one fixed point. They also identify ordinary and \(L^2\)-Lefschetz numbers for pointed homotopy idempotents of connected closed smooth oriented manifolds that induce the identity on the fundamental group. Corollary 21 states these consequences precisely; Section 9 explains their connection to projective Euler characteristics.

The two parts of the proof

We first describe the algebraic construction to explain the geometric statement it needs. The body of the paper begins with that geometric statement, formulated independently of group rings, and completes its proof before constructing the cycles and cocycles to which it applies.

For a finitely generated group \(G\), the algebraic construction starts with an idempotent \(e\in M_q(\mathbb CG)\) and one infinite-order element \(g\in G\). The finitely many nonzero group coefficients of \(e\) determine a finite set of steps. Products of matrix coefficients along paths from \(h\) to \(gh\), followed by the ordinary matrix trace, give complex weights on finite lists of group elements. Idempotency says that summing over any intermediate vertex removes that vertex. Cyclicity of matrix trace gives a second identity, a cyclic shift whose last vertex is multiplied by \(g\).

Project the vertices to the coset set \(Y=\langle g\rangle\backslash G\). The group \(H=C_G(g)/\langle g\rangle\), where \(C_G(g)\) is the centralizer of \(g\), acts freely on \(Y\). For every positive even dimension \(2m\), the weighted lists define a simplicial cycle modulo this action. Its simplices use connected sets of vertices in one graph of bounded degree, fixed as \(m\) increases. A Pfaffian cocycle evaluates on this cycle as \(2^{-m}\mathop{\mathrm{HS}}_G([e(\mathbb CG)^q])([g]_G)\). Here a cocycle means an alternating function on vertex lists that vanishes on simplicial boundaries. The explicit construction and the evaluation occupy Sections 7 and 8.

The geometric part, proved first in Sections 2–6, shows that every invariant cocycle evaluates to zero on cycles of this kind in sufficiently high dimension. We cut the supporting simplicial complex successively into pieces of bounded diameter, choosing the separating sets nearly to minimize their areas. A volume estimate and repeated slicing make the expected number of points in the final zero-dimensional set arbitrarily small. A subdivision construction turns this point count into a bound for cocycle evaluations. Equivariant chain homotopy keeps the evaluation unchanged throughout the construction.

The fixed graph degree makes increasing the dimension useful. Through each vertex there are only exponentially many connected vertex sets of a given size. On an \(n\)-simplex, Euclidean structures chosen from spanning trees of its vertex set give volume \(1/n!\); repeated slicing produces the same factorial in the comparison with a point of the final separating set. After these factors cancel, radii chosen in terms of the fixed degree bound make the remaining volume ratio tend to zero with \(n\). The local replacement argument turns this strict volume margin into the small expected point count. No homological-dimension bound on \(H\) is assumed.

Two features permit this argument for arbitrary groups. A compact probability space with a free action separates overlapping translates of local modifications, without any amenability assumption. Also, each top simplex has its own averaged Euclidean area functional. The areas need not match along common faces; the successive cuts keep the relevant dimensional measures away from those faces. The cocycles need not be globally bounded: near each original simplex, the construction uses only a fixed finite set of possible vertices. Bounding the cocycle on those lists suffices for the evaluation estimate.

The final assembly in Section 9 chooses a large even dimension, forcing the individual trace coefficient to vanish. Passing from finitely generated groups to arbitrary groups uses the exact sum over subgroup conjugacy classes contained in an ambient conjugacy class. Section 10 then extracts the augmentation rank from the identity coefficient. In the scalar case this rank is \(0\) or \(1\), and the standard faithful-trace argument forces the idempotent itself to be \(0\) or \(1\). A finite-coefficient reduction extends that conclusion from complex coefficients to every commutative characteristic-zero domain.

Relation to previous work

The rank construction comes from Hattori and Stallings (Hattori 1965; Stallings 1965); Bass developed its relation to Euler characteristics and formulated the integral conjecture in (Bass 1976); see also (Bass 1979, Conjecture (4.5)). Bass proved complex trace vanishing at infinite-order elements for linear groups (Bass 1979, Theorem (6.1)(f)). For the integral trace, Linnell established a restriction on the possible nonzero coefficients (Linnell 1983, Lemma 4.1), later recovered through topological Hochschild homology by Berrick and Hesselholt (Berrick and Hesselholt 2015, Theorem A). Its finite-order consequence is the input used in Section 9.

The scalar idempotent problem also has a long history of trace methods. Formanek proved the characteristic-zero conclusion for torsion-free groups satisfying the ascending chain condition on subgroups (Formanek 1973, Theorem 1). Burger and Valette revisit the theorems of Kaplansky and Zalesskii: the canonical trace of a complex scalar idempotent lies in \([0,1]\), its endpoints force the idempotent to be trivial, and its value is rational (Burger and Valette 1998, Theorems 1.3 and 2.1). They also obtain \(\tau_{G,*}(K_0(\mathbb CG))\subseteq\mathbb Q\) (Burger and Valette 1998, Corollary 1.2). Rationality still permits nonintegral values strictly between zero and one. For torsion-free groups, Theorem 1 identifies the trace with augmentation rank, which excludes those values. The subsequent faithful-trace argument is classical and is included in Section 10.

Broader group classes have been treated by algebraic and analytic methods. Farrell and Linnell proved the complex Bass statement for elementary amenable groups as part of a result over integral domains (Farrell and Linnell 2003, Theorem 1.6). Berrick, Chatterji and Mislin obtained it for amenable groups through rational surjectivity of the degree-zero Bost assembly map for countable groups (Berrick et al. 2004, Theorems 1.3–1.4 and Lemma 1.5). Their result concerns the stronger trace assertion on \(\ell^1(G)\), with finite support on finite-order classes. An algebraic route uses induction from finite subgroups: the Farrell–Jones conjecture with complex coefficients implies the complex Bass statement (Bartels et al. 2008, Theorems 1.5 and 1.9). The present proof uses the finite support of group-ring idempotents throughout; it requires neither an assembly hypothesis nor a statement about a completed group algebra.

For some groups, analytic methods give a stronger idempotent conclusion in the reduced group \(C^*\)-algebra \(C_r^*(G)\), the operator-norm closure of \(\mathbb CG\) acting on \(\ell^2(G)\). Using Lafforgue’s Banach \(KK\)-theory, Mineyev and Yu proved this for torsion-free subgroups of hyperbolic groups (Mineyev and Yu 2002, Theorems 19–21). Puschnigg independently obtained the torsion-free hyperbolic case through local cyclic homology (Puschnigg 2002); see also (Mineyev and Yu 2002, sec. 6). Our conclusion for arbitrary torsion-free groups concerns the group ring and the trace on its algebraic \(K_0\), as distinguished in Section 10.

The higher-dimensional algebraic construction has antecedents in the cyclic-homological approach to the conjecture. Connes constructed a pairing of even cyclic cocycles with projective classes (Connes 1985, pt. II, §2, Proposition 14). Burghelea’s decomposition of cyclic homology by conjugacy classes identifies the infinite-order components through the reduced centralizers \(C_G(g)/\langle g\rangle\) (Burghelea 1985, Proposition 1.8 and Theorem I). Eckmann used higher-degree vanishing to obtain trace vanishing over rational group rings for specified classes of groups of finite rational homological dimension (Eckmann 1986, sec. 3, Theorems 3.1 and 3.3). Sections 7–8 give direct finite-chain constructions in this tradition, including the normalization of the Pfaffian detector; no cyclic-homology theorem is needed as an input. The sparse-chain theorem replaces a homological-dimension hypothesis with connected-support control on the particular cycles.

The geometric proof develops the separating-set method of Papasoglu (Papasoglu 2020, sec. 2, especially Lemma 2.5), the separating filtrations and simplex-counting argument of Alpert (Alpert 2024, Lemmas 4, 6–7), and the Cantor-parameter approach of Braun and Sauer (Braun and Sauer 2021, secs. 2, 4–5). Alpert’s equivariant formulations (Alpert 2024, sec. 4, especially Lemmas 11–12) provide the closest model for the local comparisons. Alpert’s filtration results concern manifold geometry and compatible metrics. We give the full piecewise-linear argument needed for simplex-dependent areas and arbitrary cycles in the coinvariant chain complex. The separate Pfaffian calculation converts this geometric vanishing into a statement about each conjugacy-class coefficient.

The metric and its finite local geometry

Represent a point of \(|K|\) by a probability vector \(p\) on \(Y\) whose support lies in a simplex of \(K\). Let \(d_Y\) denote graph distance and define \[ \rho(p,q)=\frac1{\sqrt n} \sup_{\operatorname{Lip}_{d_Y}(\ell)\leq1} \sum_{v\in Y}\ell(v)\bigl(p(v)-q(v)\bigr). \tag{3}\] This is the scaled Kantorovich–Rubinstein metric on finitely supported probability measures, in its Lipschitz-potential form (Kantorovich 2006, 1381–82). We establish its needed properties directly. Only finitely many terms occur in the pairing. To see that the supremum is finite and describe the distance function on each simplex, let \(V\subset Y\) be a nonempty finite set and put \[\mathcal P_V= \left\{\ell\in\mathbb R^V: \sum_{v\in V}\ell(v)=0,\quad |\ell(v)-\ell(w)|\leq d_Y(v,w)\quad(v,w\in V)\right\}.\] This is a nonempty compact polytope. Indeed, its difference constraints and zero mean bound every coordinate by \(\max_{v,w\in V}d_Y(v,w)\). Each \(\ell\in\mathcal P_V\) extends to a \(1\)-Lipschitz function on \(Y\) by \[\widetilde\ell(y)=\min_{v\in V}\bigl(\ell(v)+d_Y(v,y)\bigr).\] The difference constraints show that this extension agrees with \(\ell\) on \(V\), and the triangle inequality proves that it is \(1\)-Lipschitz. Conversely, a global \(1\)-Lipschitz function restricts, after subtraction of its mean, to an element of \(\mathcal P_V\). Constants pair to zero with \(p-q\). Consequently, when \(V\) contains both supports, the supremum in (3) is the maximum over \(\mathcal P_V\).

Symmetry follows by replacing \(\ell\) with \(-\ell\), and the triangle inequality follows by applying each potential to \(p-q=(p-u)+(u-q)\). The indicator of a single vertex is \(1\)-Lipschitz for \(d_Y\), so \[ |p(v)-q(v)|\leq\sqrt n\,\rho(p,q) \qquad(v\in Y). \tag{4}\] Thus \(\rho\) separates points and is a metric. It is \(H\)-invariant because graph automorphisms permute the admissible potentials. On probability vectors supported in a fixed finite set, it is the restriction of a norm on the finite-dimensional space of vectors with coordinate sum zero. It therefore gives the usual topology on every finite subcomplex.

For \(c\in|K|\), write \[\begin{aligned} f_c(p)&=\rho(c,p),& U_s(c)&=\{p:f_c(p)<s\},\\ L_s(c)&=\{p:f_c(p)\leq s\},& S_s(c)&=\{p:f_c(p)=s\}. \end{aligned}\] On a fixed closed simplex \(\sigma\), the function \(f_c\) is a maximum of finitely many affine functions: use the extreme points of \(\mathcal P_{V(\sigma)\cup\mathop{\mathrm{supp}}c}\). Here \(V(\sigma)\) denotes the vertex set of \(\sigma\). Taking all these extreme points makes the collection of affine branches equivariant under \(H\).

We will also use the following consequence. For fixed \(c\) and \(s\), only finitely many \(h\in H\) satisfy \[ L_s(hc)\cap L_s(c)\ne\varnothing. \tag{6}\] Indeed, an intersection gives \(\rho(hc,c)\leq2s\), and then (5) puts some vertex of \(h\mathop{\mathrm{supp}}c\) within distance \(2s\sqrt n\) of \(\mathop{\mathrm{supp}}c\). The same injectivity argument applies.

Simplex areas and distance gradients

The metric now provides finite neighborhoods in which we can make cuts. For area comparisons, we use Euclidean structures separately inside the top simplices. They need not agree on shared faces.

Fix a top simplex \(\sigma\). For each spanning tree \(T\) of the connected graph on \(V(\sigma)\), equip its affine span with the Euclidean structure in which the tree-edge vectors \(\delta_v-\delta_w\), with either orientation for each edge, form an orthonormal basis of the translation space. For \(0\leq j\leq n\), let \(a_j^\sigma\) be the average, over these finitely many trees, of Euclidean \(j\)-area on polyhedral subsets of \(\sigma^\circ\) of dimension at most \(j\). We allow relatively open faces in these sets; pieces of dimension less than \(j\) have zero \(j\)-area, and \(a_0^\sigma\) is counting measure. These area functionals are covariant under \(H\), since the action permutes the spanning trees.

The volume margin in high dimension

We next bound the total top-dimensional area of a metric ball by counting its possible connected vertex supports. Fix \[R=4D^2,\qquad r=R/2.\] For a specified vertex \(v\), there are at most \(D^{2n}\) connected sets of \(n+1\) vertices containing \(v\). To prove this, choose a spanning tree of each such set rooted at \(v\). A traversal using each tree edge once in each direction is a walk of length \(2n\), starting at \(v\), whose visited set is exactly the prescribed set. Choosing one such walk per set gives an injection into the length-\(2n\) walks from \(v\), of which there are at most \(D^{2n}\).

Set \(k=\lfloor R\sqrt n\rfloor\). A graph ball of radius \(k\) has at most \[\sum_{j=0}^kD^j\leq D^{k+1}\] vertices, since \(D\geq2\). As \(|\mathop{\mathrm{supp}}c|\leq n+1\), there are at most \((n+1)D^{k+1}\) vertices within distance \(k\) of \(\mathop{\mathrm{supp}}c\). By (5), each top simplex meeting \(L_R(c)\) contains one of them. The connected-set count and Lemma 5 therefore give, for every \(c\in|K|\), \[ \sum_{\sigma\ \mathrm{top}} a_n^\sigma\bigl(L_R(c)\cap\sigma^\circ\bigr) \leq V_R:= \frac{(n+1)D^{\lfloor R\sqrt n\rfloor+1}D^{2n}}{n!}. \tag{7}\] Counting a simplex more than once only enlarges this upper bound.

With \(D\) fixed, the ratio of this estimate to \((R-r)^n/n!\) is \[\frac{V_R}{(R-r)^n/n!} =\frac{(n+1)D^{\lfloor4D^2\sqrt n\rfloor+1}}{2^n} \longrightarrow0.\] Indeed, its logarithm is \(-n\log2+4D^2(\log D)\sqrt n+O_D(\log n)\). We may therefore choose \(n_0(D)\geq1\) so that, for every \(n\geq n_0(D)\), \[ V_R<\frac{(R-r)^n}{n!}. \tag{8}\] For the remainder of the geometric proof, fix such an \(n\). The graph degree bound \(D\), and hence \(R\) and \(r\), remain fixed as dimensions increase. The strict margin (8) will convert successive area comparisons into a bound on the number of points at the bottom of a separating filtration.

A free probability parameter

The following construction is a finite-color variant of the coinduction argument in Braun and Sauer (Braun and Sauer 2021, Theorem 2.1), who credit the existence theorem to Hjorth and Molberg (Hjorth and Molberg 2006, Theorem 0.1). We include the construction, including its torsion case, so that no free-action theorem is an input.

Fix such \((X,\mu)\) for the rest of the geometric argument. In particular, no amenability assumption on \(H\) is needed.

The allowed filtrations

A triangulation refining \(K\) means a finite affine simplicial subdivision of each closed original simplex, with agreement on shared faces. A family \(T(x)\) of such triangulations, together with specified subcomplexes, is locally constant if, on each original simplex, all its exact affine simplices and subcomplex markings are constant on the pieces of a finite clopen partition of \(X\). The partition may depend on the original simplex. We require equivariance: \(T(hx)=hT(x)\), and likewise for every specified subcomplex. Refinement of triangulations always preserves their earlier subcomplexes.

Begin with \(T_n(x)=K\) and \(Z_n(x)=|K|\). At stage \(d\), where \(n\ge d\ge1\), suppose that \(P(x)=Z_d(x)\) is a subcomplex of \(T_d(x)\). An admissible next choice consists of a locally constant equivariant refinement \(T_{d-1}(x)\) and its subcomplex \(Z_{d-1}(x)\) such that

1. \(Z_{d-1}(x)\subseteq P(x)\);

2. for every simplex \(F\) of \(T_d(x)\) contained in \(P(x)\), \[\dim\bigl(F\cap Z_{d-1}(x)\bigr)\le\dim F-1;\]

3. every connected component of \(P(x)\setminus Z_{d-1}(x)\) has \(\rho\)-diameter at most \(2R\).

A negative dimension bound means that the intersection is empty. Thus the new separator avoids the vertices of \(T_d\) and meets each higher-dimensional face in lower dimension; its own vertices belong to the finer triangulation \(T_{d-1}\). Connected components use the subspace topology given by \(\rho\). We call a filtration satisfying these conditions admissible. In particular, these conditions imply \(\dim Z_i(x)\le i\).

The local operation behind the construction is easiest to describe at a fixed parameter. For a separator \(Z\subseteq P\), a center \(c\in|K|\), and \(t\in(r,R)\), remove the part of \(Z\) inside the ball and insert the corresponding slice of \(P\): \[Z^{\prime}=(Z\setminus U_t(c))\cup(P\cap S_t(c)).\] The inserted slice prevents a component of \(P\setminus Z^{\prime}\) from passing between the ball and its exterior. Inside the ball its diameter is at most \(2t\); outside, the old separator remains. The next lemma makes this operation compatible with triangulations, group translates, and clopen parameter choices. It also constructs the level cuts used to start a filtration. Only the affine descriptions of distance levels enter the construction; no Euclidean metrics need to agree across faces.

The genericity against \(S\) is useful when \(S\) already contains the old separator \(Z\): intersections of an inserted level with its faces have one less dimension. The later area comparison will use precisely this conclusion. The lemma itself does not require disjoint active balls; disjointness will be imposed when their areas are counted.

Dimension drop along the entire filtration

We record the consequence that will control chains in Section 6. For any simplex \(F\) of \(T_i(x)\) contained in \(Z_i(x)\) and any \(0\le j\le i\), \[ \dim\bigl(F\cap Z_j(x)\bigr)\le\dim F-(i-j). \tag{10}\] Indeed, the first step is the defining condition. Its intersection is a subcomplex in the next refinement. Applying the next dimension drop to each simplex of that subcomplex decreases its maximal dimension by one. Induction gives (10). Applied with \(i=n\), this also shows that a proper original face meets \(Z_j(x)\) in dimension strictly less than \(j\).

We have obtained a nonempty class of equivariant separating filtrations, together with a local cutting operation that stays within that class. The next section chooses separators with area close to the infimum and compares them with these cuts.

Successive near minimizers

The comparison with a cut through a ball follows the separating-set method of Papasoglu (Papasoglu 2020, sec. 2, especially Lemma 2.5) and its iterated and equivariant forms in Alpert (Alpert 2024, Lemmas 6–7 and 11–12). We prove the comparisons for our simplex-dependent areas, retaining the error from each near minimizer.

Component averages and their finite support

For a fixed parameter \(x\), take the barycentric subdivision of the final triangulation \(T_0(x)\). Its vertices are the barycenters \(b_F\) of nonempty simplices \(F\) of \(T_0(x)\). Define \[\ell(F)=\min\{i:b_F\in Z_i(x)\}, \qquad Z_{-1}(x)=\varnothing.\] All \(Z_i(x)\) are subcomplexes of \(T_0(x)\), so \(b_F\in Z_i(x)\) is equivalent to \(F\subseteq Z_i(x)\). Let \(U_F\) be the connected component of \(Z_{\ell(F)}(x)\setminus Z_{\ell(F)-1}(x)\) containing \(b_F\). For any such component \(U\), set \[ J_U=\bigcup_{p\in U}\mathop{\mathrm{supp}}p, \qquad u_U=\frac{1}{|J_U|}\sum_{v\in J_U}\delta_v. \tag{21}\] The next lemma proves that \(J_U\) is finite and controls these labels as \(x\) varies. The intended image of an oriented barycentric simplex is \[ [b_{F_0},\ldots,b_{F_j}] \longmapsto u_{U_{F_0}}\wedge\cdots\wedge u_{U_{F_j}}\in E_j. \tag{22}\]

This choice uses the separating filtration in a specific way. If nested faces \(F\subset F'\) have the same level \(i\), the segment joining their barycenters lies in \(Z_i\setminus Z_{i-1}\). Except at its endpoint \(b_F\), it lies in the relative interior of \(F'\), which the subcomplex \(Z_{i-1}\) misses; the endpoint also avoids \(Z_{i-1}\) by its level. The two barycenters therefore receive the same component vector, making any exterior term containing both zero. Along a complete flag \(F_0\subset\cdots\subset F_n\), with \(\dim F_i=i\), the levels are nondecreasing: if \(F\subset F'\subseteq Z_i\), then \(F\subseteq Z_i\). A nonzero image must consequently use all levels \(0,1,\ldots,n\), beginning at a point of \(Z_0\). We will bound the number of these flags after establishing that the labels define finite chains.

Subdivide each oriented original simplex into its fundamental subdivision chain and apply the exterior assignment (22) to every term. This defines a map \(f_x:C_*(K)\to E_*\). Both operations commute with the augmented boundary. For subdivision, interior facets cancel in pairs and the boundary facets give the compatible subdivision on each original face. For the second operation, each assigned vector has coefficient sum one, so contraction with the coefficient-sum functional gives \[\partial(u_0\wedge\cdots\wedge u_j) =\sum_{i=0}^j(-1)^i u_0\wedge\cdots\wedge\widehat{u_i}\wedge\cdots\wedge u_j.\] Thus \(f_x\) is an augmented chain map.

By Lemma 11, its value on any original simplex is constant on a finite clopen partition and is supported in a fixed finite exterior space. We may therefore define \[ F^\epsilon(c)=\int_X f_x(c)\,d\mu(x). \tag{23}\] This integral is an actual finite chain: for every \(c\) it is a finite linear combination of finitely many chain values. It commutes with the augmented boundary. Moreover, \(f_{hx}(hc)=h f_x(c)\), so invariance of \(\mu\) shows that \(F^\epsilon\) is \(H\)-equivariant.

Counting the surviving flags

We now bound the evaluation of \(f_x\) on a top simplex. The number of subdivision simplices may depend on \(\epsilon\); the following bound depends instead on the number of zero-stratum points. The component-labeling and flag-counting method is modeled on Alpert (Alpert 2024, Lemmas 4 and 10); the carrier argument below supplies the bound without a manifold hypothesis.

Figure 1 illustrates the two choices at each of the two stages when \(n=2\).

Let \(\psi:E_n\to\mathbb C\) be the invariant cocycle in Theorem 3. For an original top simplex \(\sigma\), put \[M_\sigma= \max_{v_0,\ldots,v_n\in W_\sigma} \left|\psi(\delta_{v_0}\wedge\cdots\wedge\delta_{v_n})\right|.\] This is finite, even though \(\psi\) need not be bounded on all basis simplices of the infinite vertex set. Every vector in (21) is a nonnegative convex combination of vertices in \(W_\sigma\). Multilinearity therefore bounds the evaluation of each surviving term by \(M_\sigma\). The preceding lemma and Proposition 10 give \[\begin{align*} \left|\psi\bigl(F^\epsilon(\sigma)\bigr)\right| &\leq 2^n M_\sigma \int_X\#\bigl(Z_0(x)\cap\sigma\bigr)\,d\mu(x) \\ &\leq 2^n M_\sigma C_\sigma\epsilon \longrightarrow 0. \tag{24}\end{align*}\] All constants here are independent of \(\epsilon\). Since chains have finite support, the same convergence holds on every fixed \(c\in C_n(K)\).

Equivariant comparison and passage to coinvariants

It remains to show that the small evaluations just obtained equal the original cocycle evaluation on every coinvariant cycle. We do this by constructing a homotopy before taking coinvariants.

A fixed graph and finite path sums

For the rest of this section, fix a finitely generated group \(G\), an idempotent \(e\in M_r(\mathbb CG)\), and an infinite-order element \(g\in G\). Write \[e=\sum_{s\in G}A_s s,\qquad A_s\in M_r(\mathbb C),\qquad S_0=\{s\in G:A_s\ne0\}.\] The set \(S_0\) is finite. Idempotence says that \(\sum_{uv=t}A_uA_v=A_t\) for every \(t\in G\). Thus the matrix-valued kernel \[a(h,h')=A_{h^{-1}h'}\] is invariant under simultaneous left translation and satisfies \[ \sum_{k\in G}a(h,k)a(k,h') =\sum_{uv=h^{-1}h'}A_uA_v =a(h,h'). \tag{26}\] All these sums are finite. In particular, the order of the matrix factors agrees with the column convention for right modules.

Put \[J=C_G(g),\qquad H=J/\langle g\rangle,\qquad Y=\langle g\rangle\backslash G, \qquad \bar h=\langle g\rangle h.\] Choose a finite symmetric generating set \(S\subset G\setminus\{1\}\) containing \(S_0\setminus\{1\}\) and its inverses. Give \(Y\) the simple undirected graph whose edges join \(\bar h\) to \(\overline{hs}\) for \(s\in S\); loops are omitted and repeated edges identified. The graph is connected and has degree at most the fixed integer \[D=\max\{2,|S|\}.\] The group \(H\) acts by left multiplication. This is well-defined because \(J\) centralizes \(g\), and it preserves right-step edges. It is free on vertices: \(\overline{ch}=\bar h\) with \(c\in J\) implies \(ch=g^q h\) for an integer \(q\), and hence \(c=g^q\). Both \(H\) and \(Y\) are countable. We keep this graph and this value of \(D\) throughout all subsequent choices of dimension.

For \(k\ge1\), let \(Q_k=J\backslash G^k\), with simultaneous left translation on tuples. Define the weight of a tuple by \[ w_k(h_0,\ldots,h_{k-1}) =\mathop{\mathrm{tr}}\bigl(a(h_0,h_1)\cdots a(h_{k-2},h_{k-1})a(h_{k-1},gh_0)\bigr), \tag{27}\] where \(w_1(h_0)=\mathop{\mathrm{tr}}(a(h_0,gh_0))\). These weights are \(J\)-invariant. They define a finite complex linear combination \(v_k\) of elements of \(Q_k\). Indeed, a nonzero weight has increments \[s_i=h_i^{-1}h_{i+1}\quad(0\le i<k-1),\qquad s_{k-1}=h_{k-1}^{-1}gh_0\] in \(S_0\), with \[s_0\cdots s_{k-1}=h_0^{-1}gh_0.\] An increment list determines the tuple from \(h_0\). Any two choices of \(h_0\) with this conjugate differ by left multiplication in \(J\), so the list determines at most one element of \(Q_k\). Hence \(v_k\) has at most \(|S_0|^k\) nonzero coefficients. For a \(J\)-invariant function \(p\) on \(G^k\), write \(v_k(p)\) for the finite weighted sum of its values. These weights need not be real or positive.

Even-dimensional cycles

The path identities now turn the finite sums into cycles. We use the exterior complex \(E_*\) on the graph \(Y\). For \(k\ge1\), define \[\pi_k:\mathbb C^{(Q_k)}\longrightarrow(E_{k-1})_H,\qquad [h_0,\ldots,h_{k-1}]\longmapsto [\delta_{\bar h_0}\wedge\cdots\wedge\delta_{\bar h_{k-1}}].\] The map is well-defined because changing a tuple by an element of \(J\) changes its projected vertices by a simultaneous \(H\)-translation. Set \(b_k=\pi_k(v_k)\), and, for \(n=2m\ge2\), set \[z_n=b_{n+1}\in(E_n)_H.\]

We have obtained the required sparse cycles in arbitrarily large even degrees. The underlying path sums all have total weight \(\lambda\), the trace coefficient in (28). The next section constructs a cocycle whose value on \(z_{2m}\) recovers a nonzero scalar multiple of that coefficient.

Relative exponents between lifts

The function \(\beta\) itself need not descend to \(Y\times Y\). In fact, (29) gives \(\beta(h,g^k h)=k\) even though these two lifts project to the same vertex. Nor is there a difficulty if an element \(\bar c\in H\) exchanges two distinct projected vertices. If a lift \(c\in J\) satisfies \(c h_0=g^a h_1\) and \(c h_1=g^b h_0\), its invariance merely requires \[2\beta(h_0,h_1)=b-a,\] which is precisely what antisymmetrization gives. Real values are useful here: for \(G=\langle t\rangle\) and \(g=t^2\), the value \(\beta(1,t)=1/2\) is forced.

The cyclic Pfaffian identity

In degree two, a lift-independent expression is \[c(y_0,y_1,y_2) =\beta(h_1,h_2)-\beta(h_0,h_2)+\beta(h_0,h_1), \qquad \bar h_i=y_i.\] Changing \(h_i\) to \(g^{r_i}h_i\) adds \((r_2-r_1)-(r_2-r_0)+(r_1-r_0)=0\), so \(c\) depends only on the projected vertices. In higher even degrees we replace \(\beta\) by a Pfaffian and take the same alternating deletion sum. The Pfaffian still depends on the lifts; its deletion sum will define the cocycle. We first compute the weighted Pfaffian sum, then prove the descent and cocycle properties of its deletion sum.

For a list \(\mathbf h=(h_0,\ldots,h_{2m-1})\) with \(m\ge1\), let \[p_m(\mathbf h)=\operatorname{Pf} \bigl(\beta(h_i,h_j)\bigr)_{0\le i,j<2m}, \qquad p_0=1.\] We use the convention that \(p_m(\mathbf h)\) is the coefficient of \(\varepsilon_0\wedge\cdots\wedge\varepsilon_{2m-1}\) in \(b^m/m!\), where \(\varepsilon_i\) are the coordinate one-forms on the space of list positions and \[b=\sum_{i<j}\beta(h_i,h_j)\, \varepsilon_i\wedge\varepsilon_j.\] In particular, \(p_1(h_0,h_1)=\beta(h_0,h_1)\), and \(p_m\) is alternating in its entries and \(J\)-invariant. The finite weighted sums \[q_m=v_{2m}(p_m)\quad(m\ge1),\qquad q_0=\lambda\] are therefore defined. No boundedness of \(p_m\) on all lists is needed.

Descent to an exterior cocycle

The preceding calculation detects the coefficient on lists of lifts. We next descend it to the exterior complex on \(Y\), where the sparse-chain theorem applies.

The integral trace

For the integral group ring, the same idempotent formula defines \[\mathop{\mathrm{HS}}_{\mathbb ZG}:K_0(\mathbb ZG)\longrightarrow \mathbb ZG/[\mathbb ZG,\mathbb ZG]\cong\bigoplus_{C\in\mathop{\mathrm{con}}(G)}\mathbb ZC,\] where commutators are now taken with their additive span. The integral Bass trace conjecture asks for support only at the identity class.

Homotopy idempotents and Lefschetz numbers

The integral support statement also has fixed-point consequences. A self-map \(f\) is a homotopy idempotent if \(f\circ f\simeq f\). Such a map on a connected finite CW complex \(X\) admits maps \(u:X\to Y\) and \(v:Y\to X\) with \[v\circ u\simeq f,\qquad u\circ v\simeq\mathop{\mathrm{id}}_Y\] (Berrick et al. 2007, sec. 3). The second identity says that \(Y\) is finitely dominated by \(X\). For a pointed homotopy idempotent with \(f_\#=\mathop{\mathrm{id}}_G\), where \(G=\pi_1(X)\), the induced maps \(u_\#\) and \(v_\#\) identify \(\pi_1(Y)\) with \(G\). The universal-cover chains of \(Y\) are chain homotopy equivalent to a finite complex \(P_*\) of finitely generated projective \(\mathbb ZG\)-modules. The sum \(\sum_i(-1)^i[P_i]\in K_0(\mathbb ZG)\) is its projective Euler class. The Hattori–Stallings trace of this class records the fixed-point class indices of \(f\): it is the Reidemeister trace, with conjugacy classes inverted when passing from the left-module convention of (Berrick et al. 2007, Lemma 4.1) to ours. The augmentation trace and the identity coefficient of this projective class give the ordinary and \(L^2\)-Lefschetz numbers, respectively (Berrick et al. 2007, Lemmas 8.1–8.2). Berrick, Chatterji and Mislin use these relations to deduce the following consequences of the integral support statement and its numerical trace consequence.