Proof. The restricted map \(W\to Z\) is the connected covering corresponding to \[\ker\bigl(\pi_1(Z)\longrightarrow\pi_1(X)\bigr).\] Thus it has infinitely many sheets. A fiber is a closed discrete infinite subset, so \(W\) cannot be compact. It is closed in \(\widetilde X\), since it is a connected component of the closed inverse image of \(Z\).

Each rational component of \(Z\) is simply connected. Its inverse image therefore consists of compact rational curves mapping isomorphically onto it. Choose an evenly covered neighborhood in \(X\) of each point of \(Z\), small enough that it meets \(Z\) in one smooth branch or in the two branches of a node. Each lifted neighborhood meets at most one or two lifted components. These neighborhoods show that the collection is locally finite in the ambient manifold \(\widetilde X\); points outside the closed inverse image have a neighborhood meeting none of it. The connected space \(W\) is locally path connected; a path in it has compact image and meets only finitely many of these curves. The maximum principle makes a holomorphic function constant on each compact curve, and the constants agree at their intersections. They therefore agree throughout \(W\).

For \(w\in W\), this proves \(W\subset\widehat{\{w\}}_{\widetilde X}\). A compact hull cannot contain the closed noncompact set \(W\). ◻

Proof. The hypothesis says that \(\alpha(U_i)-U_i\) has degree at least \(w_i+a\). Expand the difference between a monomial and its substituted image. Every summand replaces at least one factor by an error of at least \(a\) additional degrees. The estimate follows for polynomials and then for series by completeness. For operator substitution, a word of weighted degree \(n\) raises degree by at least \(n\); hence only finitely many words contribute in each finite truncation. Multiplication and composition respect these substitutions. ◻

Proof. Let \(I\subset A\) be the closed two-sided ideal generated by \(f_\lambda=r_\lambda-1\), and let \(R=A/I\) with its quotient filtration. Put \[C(n)=\dim_{\mathbb F_2}R/R_{\ge n}\quad(n\ge1), \qquad C(n)=0\quad(n\le0).\] First-letter decomposition gives a surjective map \[\bigoplus_i R/R_{\ge n-w_i}\longrightarrow R_{>0}/R_{\ge n},\qquad (b_i)_i\longmapsto\sum_i U_i b_i.\] Its kernel has dimension at most \(\sum_\lambda C(n-v_\lambda)\). Here is the filtered justification. Write \(f_\lambda=\sum_i U_i f_{i\lambda}\). A vector in the kernel can be lifted to series whose first-letter sum lies in \(I+A_{\ge n}\). Subtracting the first-letter decomposition of the degree-at-least-\(n\) term does not change the truncated vector, so the sum may be taken in \(I\). For a product \(a f_\lambda h\), the positive-degree part of \(a\) contributes first-letter coefficients that vanish in \(R\); its constant part contributes \[(\overline{f_{i\lambda}})_i\,\bar h.\] This vector has shifted degree at least \(v_\lambda\). Only \(\bar h\) modulo \(R_{\ge n-v_\lambda}\) matters. These vectors therefore span a space of dimension at most \(\sum_\lambda C(n-v_\lambda)\). Closure adds nothing to a span in a finite-dimensional truncation. Since \(\dim R_{>0}/R_{\ge n}=C(n)-1\), we obtain \[ C(n)-\sum_i C(n-w_i)+\sum_\lambda C(n-v_\lambda)\ge1. \tag{3}\]

Suppose the stated pro-2 quotient were finite, of order \(m\). Its image spans every truncated algebra \(R/R_{\ge n}\): the variables are the images of \(g_i-1\), and their products are linear combinations of group elements. The group map to each truncated unit group kills the relators, hence factors through that quotient. Thus \(C(n)\le m\) for all \(n\). Multiply (3) by \(t^{n-1}\) and sum over \(n\ge1\). Boundedness of \(C(n)\) and convergence of the relator series justify the sums. This gives \[\left(1-\sum_i t^{w_i}+\sum_\lambda t^{v_\lambda}\right) \sum_{n\ge1}C(n)t^{n-1}\ge\frac1{1-t},\] contradicting (2). ◻

Proof. The group \(\ker(1-\omega)\) has order three. Its two nonzero points are exchanged by \(\iota\), and its intersection with \(E[2]\) is \(\{0\}\). This gives the five excluded values. A fixed point of \(\bar\omega\) lifts to a point satisfying \(\omega z=\pm z\); since \(1+\omega\) is a unit in \(\mathbb Z[\omega]\), all such values are excluded by (4). The three targets in (5) are therefore distinct. They avoid \(q(E[2])\), which contains the critical values of the degree-16 map \(f\). Also \(f(q(0))=q(0)\), so \(q(0)\) gives a section of the complement.

The three points \(e_j=\omega^j e_0\) are the distinct nonzero elements of \(E[2]\). A point \(p\in E\) with \([4]p=e_j\) is not two-torsion. Consequently \(q\) is unramified at \(p\), \([4]\) is étale there, and \(q\) is simply ramified at \(e_j\). Thus \(f\) has simple ramification at \(q(p)\). The sixteen points above each \(e_j\) are paired freely by \(\iota\), giving eight collisions for each \(j\) and 24 in total. ◻

Proof. Two extensions of the same point motion differ by an isotopy fixing the initial punctures and zero. Their endpoint actions therefore agree. The disk construction also extends homotopies of parameter paths. With loop composition chosen to agree with composition of the endpoint actions, uncorrected monodromy is a homomorphism. All its maps commute with \(\iota\), and \(\epsilon\) is a character, so the corrected actions also compose.

The uncorrected endpoint acts trivially on the homology of the closed torus; the correction acts there by a sign. It therefore preserves the handle-coordinate kernel \(K\) and lifts uniquely through \([4]\) fixing zero. The lift commutes with \(\iota\) by uniqueness: both compositions lift the same map and fix zero. Requiring this fixed basepoint also excludes any nontrivial deck translation. Write \(\widetilde H_u\) for the lift of \(H_u\) through \([4]\) starting at the identity; it fixes zero and gives the actual motion of the \(96\) punctures upstairs. The zero-fixed lift of the corrected endpoint is \(\iota^{\epsilon(m)}\widetilde H_1\), because \([4]\) commutes with sign. After quotienting by sign, this endpoint is the endpoint induced by \(\widetilde H_u\), hence gives the actual based sphere monodromy. This proves (8).

For surjectivity, represent a loop in \(S\) based at \(q(0)\) by a path which avoids the four branch values of \(q\) except at its endpoints. Its open interval lifts through the ordinary two-sheeted cover away from those values. Both lifted endpoints tend to zero, the unique point above \(q(0)\). Thus the lift is a based loop in the source punctured torus, as required. ◻

Proof. The vector space \(V\) over \(\mathbb F_2\) has the five basis classes \(x,y,a_0,a_1,a_2\). Two characters of \(F\) are obtained from the handle coordinates modulo two. Uncorrected monodromy preserves them, and \(\iota\) preserves them modulo two.

Choose an affine coordinate on the quotient sphere with \(q(0)=\infty\), using the same letter \(q\) for its pullback to \(E\). It is an even meromorphic function with a double pole at zero. Normalize its leading coefficient to one in an additive local coordinate \(v\) at zero, so that \(v(\iota z)=-v(z)\) and \(q(z)=v(z)^{-2}+O(1)\). For \(j=0,1,2\), consider over the moving torus complement the cover \[ Y_j^2=q(z)-\bar\omega^j s. \tag{9}\] Here \(s\) and \(\bar\omega^j s\) are expressed in the chosen affine coordinate. The right side has simple zeros precisely at the two removed \(j\)-punctures. The cover is unramified elsewhere, including at zero: with \(\eta=vY_j\), its local equation is \[\eta^2=v^2(q(z)-\bar\omega^j s)=1+O(v^2).\] Thus the points \(\eta=1\) and \(\eta=-1\) over zero give two global lifts of the fixed basepoint section throughout \(B\).

Let \(\chi_j:F\to\mathbb F_2\) be the character of this double cover on the fiber over \(s_0\). A cover of the entire family with a lifted basepoint section gives an extension of \(\chi_j\) to the fundamental group of its total space. Conjugating a fiber loop by the section loop does not change its value in \(\mathbb F_2\). Hence uncorrected based monodromy preserves \(\chi_j\). Sign has the lift \[(z,Y_j)\longmapsto(-z,-Y_j).\] This lift fixes each selected sheet over zero, because both \(v\) and \(Y_j\) change sign and \(\eta=vY_j\) does not. It follows that \(\iota_*\) also preserves \(\chi_j\), and therefore so does the corrected action \(\varphi_m\).

The character \(\chi_j\) takes value one on \(a_j\) and \(a_j^-\) and zero on the other puncture meridians. In particular it vanishes on all \(c_i\) and factors through \(V\). The three characters \(\chi_j\), together with the two handle characters, form a basis of \(V^*\): evaluation on the five displayed generators has an invertible block triangular matrix. Their common kernel is therefore \(\{g:\nu_F(g)\ge2\}\). Since all five characters are preserved, this kernel is preserved and the two induced actions on \(V\) are trivial. ◻

Proof. By 7, the changes of the five weight-one generators in (6) have valuation at least two. It remains to check the two weight-two boundaries. Fix one pair, writing \(a=a_j\) and \(a^-=a_j^-\). Pureness gives \(\varphi_m(a)=waw^{-1}\) for some \(w\in F\). Choose \(d\in F\) with \(\iota_*(a)=d a^-d^{-1}\). Commutation with \(\iota_*\) yields \[\varphi_m(a^-)=w'a^-(w')^{-1},\qquad w'=\varphi_m(d)^{-1}\iota_*(w)d.\] By 7, \(w'w^{-1}\) has valuation at least two. Replacing \(w'\) by \(w\) in the conjugation of \(a^-\) therefore changes it by an element of valuation at least three. On the other hand, conjugating \(c_j=aa^-\) by \(w\) changes it by valuation at least \(1+\nu_F(c_j)\ge3\). It follows that \[\nu_F\bigl(\varphi_m(c_j)c_j^{-1}\bigr)\ge3.\] The degree-raising substitution rule of 3 now applies to the entire weighted free basis. ◻

Proof. Each sign-paired pair of torus disks projects to one two-puncture disk on the sphere, and either disk maps homeomorphically to it. The image \(q(T_{\mathrm{core}}^{(4)})\) is the sphere exterior, a sphere with \(24\) holes. Killing its boundary loops kills its fundamental group, so the quotient defining \(\Pi_{\mathrm{pair}}\) is the quotient induced by collapsing this exterior. Van Kampen identifies it with the free group on one meridian per projected disk.

The map \(q\) commutes with the two core collapses, with the basepoints in their respective cores. Upstairs there is one cyclic generator per disk, and sign identifies them in pairs by \(a_{h,j}=a_{b_j-h,j}^{-1}\). This accounts for the entire quotient: every fixed point of sign is already in the collapsed core, and either disk in each remaining pair maps homeomorphically to the corresponding sphere disk. Passing to pro-2 completions proves (13). The based collapse diagram then gives the dense map \(\theta\) and the exact identity (14).

Finally, 6 and (15) give \[\theta\beta_m p =\theta p\varphi_m|_K =\lambda\Phi\varphi_m|_K =\lambda\Phi|_K =\theta p.\] The map \(p\) is surjective, so \(\theta\beta_m=\theta\). This argument does not require the monodromy to preserve the kernel of the pair-pinching map. ◻

Proof. Choose a positive representative in the source torus for each marking. Within the \(j\)th block these representatives are indexed by \(H\). Along a parameter loop, their endpoints are obtained by a translation in \(H\) and the common sign \((-1)^{\epsilon(m)}\). The three blocks are preserved individually. Thus the joint action is contained in \(H^3\rtimes\mathbb Z/2\), with sign acting by inversion; this is a finite 2-group.

The quotient \(q\) is simply ramified at \(e_0\), so \(\gamma_*\) exchanges the two lifts of the parameter. By the local simple ramification in 5, each marking pair has equation \(w^2=t\) near its collision. Its motion under \(\gamma_*\) is a half twist and under \(\gamma_*^2\) a full twist. These motions are supported in the disjoint pair disks and fix the exterior. A full twist acts on a disk group by conjugation by its boundary; after that boundary is killed, it fixes the remaining cyclic generator. Hence the induced action on \(\Pi_{\mathrm{pair}}\) is trivial. ◻

Proof. In the Frattini quotient, the span of the generators in one block is the permutation module \[R=\mathbb F_2[H] =\mathbb F_2[\xi,\eta]/(\xi^4,\eta^4), \qquad T_1=1+\xi,\quad T_2=1+\eta.\] The images of \(u_{i\ell,j}\) are its monomial basis \(\xi^i\eta^\ell\). The Frattini basis criterion therefore gives the first assertion. We henceforth use the stated weights.

The proof has three steps: control the last difference of a cyclic shift, propagate the two translation bounds, and then treat sign. We first establish the endpoint estimate needed for a cyclic shift. Suppose \(T^4=\mathop{\mathrm{id}}\), put \(w_i=\delta_T^i(w_0)\), and assume \[ \nu_L(w_i)\ge d+i\qquad(0\le i\le3), \qquad d\ge1. \tag{17}\] Then \[ \nu_L(w_4)\ge d+4. \tag{18}\] This assertion does not assume that \(T\) preserves the filtration. To prove it, take formal free generators \(z_0,z_1,z_2,z_3\) of weights \(d,d+1,d+2,d+3\), and define the triangular automorphism \[T'(z_i)=z_{i+1}z_i\quad(0\le i<3), \qquad T'(z_3)=z_3.\] By 3, \(T'-\mathop{\mathrm{id}}\) raises algebra degree by one. In characteristic two, \[(T')^4-\mathop{\mathrm{id}}=(T'-\mathop{\mathrm{id}})^4,\] so \(T'^4(z_0)z_0^{-1}\) has valuation at least \(d+4\). Evaluation at \(z_i=w_i\) preserves this bound by (17). The positive word \(T'^3(z_0)\) contains \(z_3\) exactly once, as its first letter. In passing from the evaluated formal fourth iterate to the actual fourth iterate, the only changed substitution is \[T(w_3)=w_4w_3 \quad\hbox{in place of}\quad T'(z_3)=z_3.\] Consequently \(T^4(w_0)=w_4\,T'^4(z_0)|_{z_i=w_i}\). Since \(T^4(w_0)=w_0\), the bound for the evaluated formal iterate proves (18).

For each fixed \(\ell,j\), apply this estimate to \(w_i=u_{i\ell,j}\), \(T=T_1\), and \(d=\ell+1\). For \(i<3\), the required difference is exactly \(\delta_{T_1}(u_{i\ell,j})=u_{i+1,\ell,j}\); (18) supplies the case \(i=3\). Thus \(T_1-\mathop{\mathrm{id}}\) raises degree by one on every basis element, and therefore on the algebra.

The same argument gives the \(T_2\) bounds on the initial sequence \(u_{0\ell,j}\). They propagate along the \(T_1\)-sequences as follows. If \(T_2(z)=vz\), commutativity of the shifts gives \[ T_2(\delta_{T_1}z) =T_1(v)(\delta_{T_1}z)v^{-1}. \tag{19}\] Suppose \(z\) has assigned weight \(d\), the element \(v\) has valuation at least \(d+1\), and \(\delta_{T_1}z\) has valuation at least \(d+1\). The already proved \(T_1\) bound gives \(\nu_L(T_1(v)v^{-1})\ge d+2\). Conjugating \(\delta_{T_1}z\) by \(v\) changes it only in degree at least \(2d+2\ge d+2\). Equation (19) therefore gives the next \(T_2\) bound. Induction proves that \(T_2-\mathop{\mathrm{id}}\) raises degree by one on the full basis.

It remains to treat \(\sigma\). Powers of either shift preserve the filtration and differ from the identity by an operator raising degree by one. Since \(\sigma(a_{0,j})=T_1^{(b_j)_1}T_2^{(b_j)_2}(a_{0,j})^{-1}\), and inversion has the same linear term in characteristic two, we have \[\nu_L\bigl(\sigma(a_{0,j})a_{0,j}^{-1}\bigr)\ge2.\] We propagate this estimate along the two difference sequences. For either shift \(T\), its inverse preserves the filtration, and \[T^{-1}-T=T^{-1}(\mathop{\mathrm{id}}-T^2), \qquad T^2-\mathop{\mathrm{id}}=(T-\mathop{\mathrm{id}})^2.\] Hence, if \(\nu_L(z)\ge d\), the elements \(\delta_{T^{-1}}z\) and \(\delta_Tz\) agree modulo degree \(d+2\). If \(\sigma(z)=vz\), the identity \(\sigma T=T^{-1}\sigma\) gives \[ \sigma(\delta_Tz) =T^{-1}(v)(\delta_{T^{-1}}z)v^{-1}. \tag{20}\] When \(\nu_L(v)\ge d+1\), the inverse-shift bound and the commutator estimate show that the right side differs from \(\delta_{T^{-1}}z\) only in degree at least \(d+2\). The preceding comparison then replaces \(\delta_{T^{-1}}z\) by \(\delta_Tz\). Starting with \(a_{0,j}\), first apply this argument along \(\delta_{T_2}\) and then along \(\delta_{T_1}\). It proves that \(\sigma\) changes each basis element of weight \(w\) only in degree at least \(w+1\). A final application of 3 proves the assertion on the completed algebra. ◻

Proof. The full Magnus degree bounds are already proved. We now use the Frattini quotient to choose enough relations to impose the whole involution; generation is detected in this linear quotient. Consider one block of the Frattini quotient, identified with \(R=\mathbb F_2[\xi,\eta]/(\xi^4,\eta^4)\) as above. Give a homogeneous polynomial of degree \(n\) weight \(n+1\). Write \(b=(b_1,b_2)\) for the shift of this block. The action of \(\sigma\) on \(R\) is \[ f(\xi,\eta)\longmapsto (1+\xi)^{b_1}(1+\eta)^{b_2} f\bigl((1+\xi)^{-1}-1,(1+\eta)^{-1}-1\bigr). \tag{22}\] Here inversion of a group generator disappears in the mod-\(2\) Frattini quotient. Since \((1+\xi)^{-1}-1=\xi+\xi^2+\cdots\), expansion of (22) shows that the part of \(\sigma-\mathop{\mathrm{id}}\) raising weight by exactly one is \[ d=(\bar b_1\xi+\bar b_2\eta) +\xi^2\frac{\partial}{\partial\xi} +\eta^2\frac{\partial}{\partial\eta}, \qquad (\bar b_1,\bar b_2)=b\bmod2. \tag{23}\] The first term means multiplication.

Set \(u=\bar b_1\xi+\bar b_2\eta\), which is nonzero, and choose a linear coordinate \(v\) independent of \(u\) over \(\mathbb F_2\). The fourth-power relations become \(u^4=v^4=0\). Moreover, the vector field \(\xi^2\partial_\xi+\eta^2\partial_\eta\) sends any linear form over \(\mathbb F_2\) to its square. In these coordinates, \[ R=\mathbb F_2[u,v]/(u^4,v^4), \qquad d=u+u^2\frac{\partial}{\partial u} +v^2\frac{\partial}{\partial v}. \tag{24}\] The \(u\)-part sends \(1\) to \(u\), \(u^2\) to \(u^3\), and odd powers to zero. Its square is zero, as is the square of the \(v\)-part. The two parts commute, so \(d^2=0\).

Take the homogeneous subspace \[ C=\operatorname{span}_{\mathbb F_2} \{u^a v^b:a\in\{0,2\},\ 0\le b<4\}. \tag{25}\] On \(C\), the odd-\(u\) component of \(d\) is multiplication by \(u\), an isomorphism from \(C\) to the span of the odd-\(u\) monomials. Thus \(d|_C\) is injective and \[ R=C\oplus dC,\qquad \ker d=\mathop{\mathrm{im}}d=dC. \tag{26}\] Indeed the direct-sum assertion follows by projecting to the odd-\(u\) monomials; both summands have dimension eight. It then follows from \(d^2=0\) that the image and kernel, each of dimension eight, coincide.

For each of the eight monomials in (25), expand it in the original homogeneous monomials \(\xi^i\eta^\ell\). Choose \(g\) to be a product of the corresponding \(u_{i\ell,j}\), each appearing with its coefficient in \(\mathbb F_2\), in any fixed order. Its assigned weight \(w_g\) is the polynomial degree plus one. Then \(\nu_L(g)\ge w_g\), and 11 gives \(\nu_L(r_g)\ge w_g+1\). In the graded Frattini quotient the leading vectors of \(g,r_g\) are respectively the chosen basis of \(C\) and its image under \(d\). By (26) these form a homogeneous basis. The actual Frattini vectors therefore form a basis as well: any linear dependence would give one between their lowest nonzero weight components. Doing this in all three blocks shows that the 48 elements \(g,r_g\), equivalently \(g,\sigma(g)\), generate \(L\).

Let \(N\) be the closed normal subgroup generated by these 24 elements \(r_g\). It is invariant under \(\sigma\), because \[\sigma(r_g)=g\sigma(g)^{-1}=r_g^{-1}.\] The quotient \(L/N\) is generated by the images of the \(g\), since there \(\sigma(g)=g\); its induced involution fixes those generators and hence is the identity. Thus \(N\) is exactly the closed normal subgroup imposing \(\sigma=\mathop{\mathrm{id}}\).

Finally, in each block the chosen monomials have weight polynomial \[t(1+t^2)(1+t+t^2+t^3)=t(1+t)(1+t^2)^2.\] Adding the three blocks gives (21). Since each \(r_g\) has degree at least \(w_g+1\), its contribution to the relation bound is at most \(t^{w_g+1}\) for \(0<t<1\). ◻

Proof. Let \(A_L\) be the completed Magnus algebra of \(L\). Represent \(L\rtimes H\) on \(A_L\) by letting \(l\in L\) act by left multiplication and letting \(H\) act by its shifts. By 11, the operators representing \(x-1\) and \(y-1\) raise degree by one. So does the operator representing \(a_j-1\), which is left multiplication by \(a_{0,j}-1\). The operators representing \(c_0-1,c_1-1\) are zero, and in particular satisfy their required weight-two bounds.

Substitute these operators into the weighted Magnus algebra of \(\widehat F\). This substitution converges: a monomial of weight at least \(n\) becomes an operator raising degree by at least \(n\). Consequently the operator representing \(f-1\) raises degree by at least \(n\). If \(\Phi(f)=(l,h)\), its action on \(1\in A_L\) is \(l\), since every shift fixes \(1\). Evaluating the operator for \(f-1\) on \(1\) therefore gives \(l-1\), proving \(\nu_L(l)\ge n\). On \(\widehat K\) the \(H\)-component is trivial, which proves the last assertion. ◻

Proof. For each imposed \(m\), equality on the basis \(k_i\) gives equality of the two continuous homomorphisms on all of \(\widehat K\). The set \[E_0=\{m\in\widehat\Gamma:f_K\varphi_m=f_K \text{ on }\widehat K\}\] is a closed subgroup. For multiplication, apply one equality to \(\varphi_n(k)\) and then the other to \(k\); for inverses apply the equality to \(\varphi_m^{-1}(k)\). Closedness follows from continuity. It contains all \(\mathcal M_N\) with \(N\ge J\), and hence contains \(D_J\).

The last assertion uses the actual surjection \(K\to\Pi\) and its exact compatibility with the two based actions, proved in [prop:based-monodromy,lem:pinched-sphere]. Surjectivity onto the unpinched sphere group is essential here; no invariance of the pinching kernel was assumed in defining the relators. ◻

Proof. A main-component meridian is represented by the circuit along the fixed exterior section, so it is killed by (31).

For a tail use the blowup chart \(w=uv\). A meridian at a sufficiently large fixed slope \(v=c\) is represented downstairs by \[u=\varepsilon e^{i\theta},\qquad w=c\varepsilon e^{i\theta},\qquad 0\le\theta\le2\pi.\] Choose \(|c|\) larger than the two branch slopes and then choose \(\varepsilon\) small. Trivialize the pair motion by an isotopy supported inside the circle containing the two marks but not this meridian. In this trivialization the loop is the base circuit followed by one full pair-boundary loop, up to conjugacy and the choice of multiplication convention. The base circuit is killed by the section splitting; the pair boundary is killed by the pinching map. Both factors therefore have trivial image. ◻

Proof. Only infinitude remains to prove. The generators of \(L\) have polynomial \(P(t)\) and the involution relators have cost at most \(tG(t)\), where 12 gives \[P(t)=(1+t)G(t),\qquad G(t)=3t(1+t)(1+t^2)^2.\] At the rational value (30), \(t^2>3/50\), so \[G(t)> \frac{3\cdot49\cdot249\cdot53^2}{200^2\cdot50^2} =\frac{102817827}{100000000} >\frac{257}{250}.\] The complete weighted expression in 4 is therefore less than \[1-P(t)+tG(t)+\frac1{100}+\frac1{100} =1-G(t)+\frac1{50} <-\frac1{125}<0.\] The infinite monodromy-relator family is finite in each bounded degree and has convergent cost by (29). The remaining families are finite. All hypotheses of 4 hold, proving that \(Q\) is infinite. Passage to the quotient preserves all earlier equalities and the density of each fiber image. ◻

Proof. We first find a finite quotient detecting all local boundary images, then compactify and resolve the corresponding cover, and finally check that the original quotient map kills every new boundary meridian.

Choose a small coordinate polydisk \(V\) about a point of \(D\). Its boundary complement is \[V\setminus D\simeq(\Delta^*)^r\times\Delta^{2-r}, \qquad r\in\{1,2\}.\] After choosing an access path in \(U\), let \(A_V\subset Q\) be the image of its fundamental group. This group is generated by \(r\) commuting finite-order elements, so it is finite. Only finitely many such images occur up to conjugacy: there is one meridian type on the smooth part of each boundary component and one local group at each crossing. The smooth part of an irreducible component, with its finitely many crossings removed, is connected; moving the access path along it conjugates the local group.

For every nonidentity element in representatives of these finite groups, choose an open normal subgroup of \(Q\) that does not contain it. Their finite intersection \(N\) is open and normal. It satisfies \[ N\cap gA_Vg^{-1}=\{1\} \qquad(g\in Q) \tag{34}\] for every local group. Set \(P_f=Q/N\). Density makes the map \(\pi_1(U)\to P_f\) surjective, so its kernel defines a connected finite Galois cover \(\pi:U'\to U\).

The cover has the complex structure obtained by lifting local charts. It is algebraic and finite étale by the Riemann existence theorem for complex varieties (Grothendieck et al. 2003, Exposé XII, Théorème 5.1). This theorem applies to the nonproper surface \(U\). Normalize \(Y\) in the function field of \(U'\) and denote the resulting morphism by \(n:\overline Y\to Y\). It is finite: complex varieties are Nagata, so finiteness of relative normalization applies to the finite-type morphism \(U'\to Y\) (The Stacks Project Authors 2026, Tag 03GR). Its restriction over \(U\) is \(U'\), because the latter is already normal. Finite morphisms are projective, so \(\overline Y\) is projective (The Stacks Project Authors 2026, Tag 0B3I).

Resolve \(\overline Y\) by projective blowups, leaving its smooth locus unchanged; strong resolution in characteristic zero has this prescribed-open property (Encinas and Hauser 2002, 822–23). Next resolve the reduced boundary on the resulting smooth surface. For curves on a smooth surface this can be done by point blowups: resolve the singular component germs, decrease tangency intersection multiplicities, and separate crossings of three or more branches (The Stacks Project Authors 2026, Tag 0BIC and its proof). Choose these centers only where the boundary is not already simple normal crossings. Thus this additional operation preserves every open subset on which the pair was already smooth with simple normal crossings. We obtain \[X\xrightarrow{\ r\ }\overline Y \xrightarrow{\ n\ }Y, \qquad p=n\circ r,\] with \(X\) smooth and projective and with \(D_X\) a simple-normal-crossings divisor. The dense open set \(U'\) is unchanged, so \(X\) is connected.

We prove the factorization, including the meridians of resolution exceptional curves. The inclusion \(i\) is surjective on fundamental groups: a loop can be perturbed off a real-codimension-two divisor. Its kernel is normally generated by the component meridians of \(D_X\). Indeed, perturb a null-homotopy disk relative to its boundary so that it avoids the finitely many crossings of \(D_X\) and meets its smooth part transversely. Deleting small disks around the finitely many intersection points expresses its boundary loop as a product of conjugates of meridians and their inverses.

Let \(E\) be any component of \(D_X\), choose a smooth point \(x\in E\) away from the other components, and take a small transverse disk \(T\) through \(x\) with \(T\setminus\{x\}\subset U'\). Choose a boundary polydisk \(V\) about \(p(x)\) downstairs. By shrinking \(T\), we may assume \(p(T)\subset V\). Consequently the projection of a small meridian in \(T\setminus\{x\}\) lies in \(V\setminus D\). After including its access path, its \(Q\)-image belongs to a conjugate of \(A_V\). This argument applies also when \(E\) is exceptional and is contracted to a boundary point.

The projected meridian has trivial image in \(P_f\), since its lift is a closed loop in \(U'\). Its \(Q\)-image therefore lies in both \(N\) and a conjugate of \(A_V\), and is trivial by (34). All generators of \(\ker i_*\) are thus killed by \(\rho\circ\pi_*\). This proves (33). ◻

Proof. Every vertical meridian along \(D_*\) is trivial in \(Q\), and hence in \(P_f\), by 15. We describe the normalization locally at every kind of point of \(D_*\). The restriction of the Galois cover to a boundary-complement polydisk is classified by its local meridian homomorphism to \(P_f\); its connected components correspond to cosets of the local image.

At a smooth unmarked point of \(D_*\), the sole meridian is trivial. Every connected local cover therefore extends as a copy of the polydisk. At a node between the main component and a tail, both vertical meridians are trivial, and the same assertion holds.

At a horizontal–vertical intersection choose coordinates \((u,v)\) with the vertical curve \(u=0\) and the horizontal marking \(v=0\). The \(u\)-meridian is trivial. If the image of the \(v\)-meridian has order \(e\), the kernel of the local map \(\mathbb Z^2\to P_f\) is \(\mathbb Z\times e\mathbb Z\). Each connected local cover thus has the finite normal extension \[ (u,w)\longmapsto (u,v)=(u,w^e). \tag{35}\] This extension is smooth, and its reduced boundary is \(uw=0\). Analytification preserves finiteness and normality (Grothendieck et al. 2003, Exposé XII, Propositions 2.1(vi) and 3.2(vi)). Thus the local extensions just described agree with \(\overline Y\) by uniqueness of finite normal extension (Grothendieck et al. 2003, Exposé XII, Proposition 5.3). The entire preimage of \(D_*\) lies in the smooth simple-normal-crossings locus of the normalized pair. The resolution in 17 can be the identity on this open neighborhood.

Consider now an irreducible component above the main sphere. Its map to that sphere is unramified, including at the attaching nodes, and there are no horizontal markings on the main component. It is therefore a connected unramified cover of \(\mathbb P^1\), hence a sphere mapping with degree one. A component above a tail can branch only at its two horizontal markings, and is unramified at the attaching node. Removing those two points downstairs and all their preimages upstairs gives a connected finite cover of \(\mathbb C^*\). Its covering subgroup is \(d\mathbb Z\subset\mathbb Z\). Thus it is the covering \(z\mapsto z^d\) of \(\mathbb C^*\), whose smooth compactification is \(\mathbb P^1\).

There are no additional components from resolution over \(D_*\). The local models show that the components just identified are smooth and meet only in ordinary transverse crossings. This proves the assertion about \(Z\). ◻

Proof. Let \(F_s\subset U\) be a punctured smooth fiber, with \(s\in B'\), and choose access paths when comparing its group with \(\pi_1(U)\). Its image under \(\rho\) is dense in \(Q\) by 16. In particular its image in \(P_f\) is all of \(P_f\), so its inverse image \(F'_s\subset U'\) is connected. Under the covering inclusion, its group is the preimage of \(N\) in \(\pi_1(F_s)\). Thus the image of \(\pi_1(F'_s)\) in \(Q\) is \[\rho\bigl(\pi_1(F_s)\bigr)\cap N.\] It is dense in \(N\), because \(N\) is open and the fiber image is dense in \(Q\). The group \(N\) is infinite: a finite open subgroup would make its finite-index overgroup \(Q\) finite. Consequently the image of \(\pi_1(F'_s)\) in \(Q\) is infinite. By (33), its image in \(\pi_1(X)\) is infinite as well.

We transfer this conclusion to \(Z\) using an explicit deformation-retract neighborhood. Regard \(X\) as a compact real analytic manifold and \(Z\) as a closed semianalytic subset. This pair admits a compatible finite triangulation (Łojasiewicz 1964, sec. 3, Theorem 2, p. 464). Write it as a finite simplicial pair \((\mathcal K,\mathcal A)\), and barycentrically subdivide so that \(\mathcal A\) is a full subcomplex: any simplex whose vertices belong to \(\mathcal A\) belongs to \(\mathcal A\). For a point of \(|\mathcal K|\), let \(s_{\mathcal A}\) be the sum of its barycentric coordinates at vertices of \(\mathcal A\). On the open neighborhood \[V=\{s_{\mathcal A}>0\}\] retain those coordinates, divide them by \(s_{\mathcal A}\), and set all other coordinates to zero. Fullness ensures that the result belongs to \(|\mathcal A|\). The straight-line homotopy inside each simplex gives a strong deformation retraction \(V\to Z\). In particular the inclusion of \(V\) into \(X\) is homotopic to a map through \(Z\).

Properness of \(h\) supplies the needed containment of nearby fibers. The set \(h(X\setminus V)\) is closed in \(C'\) and does not contain \(c_*\). Hence every whole fiber over a sufficiently small neighborhood of \(c_*\) lies in \(V\). Choose \(s\in B'\) in this neighborhood. Its connected punctured lifted fiber \(F'_s\) lies in one connected component of \(V\); the deformation retraction takes that component into a connected component \(Z_0\) of \(Z\). Therefore its inclusion-induced homomorphism to \(\pi_1(X)\) factors, up to a change-of-basepoint conjugation, through \(\pi_1(Z_0)\). Since the former has infinite image, so does the latter. ◻

Proof of 1. The construction above gives a smooth connected projective surface \(X\). By [lem:rational-special-fiber,lem:infinite-special-image], it contains a connected nodal curve \(Z_0\) with smooth rational irreducible components and infinite image in \(\pi_1(X)\).

2 now applies to \((X,Z_0)\). In the simply connected universal cover \(\widetilde X\), every connected component \(W\) above \(Z_0\) is a closed noncompact locally finite union of compact rational curves. For every \(w\in W\), the holomorphic hull of \(\{w\}\) contains \(W\) and is noncompact. This proves all the assertions of 1. ◻