Historical context

Shafarevich’s question extends the uniformization viewpoint for curves to universal covers of projective varieties. Kollár’s account (Kollár 1995, Introduction, Section 0.3) explains both the holomorphic-convexity conjecture and its formulation for large fundamental groups. These formulations are related by Remmert reduction: a holomorphically convex complex manifold admits a proper holomorphic map to a Stein space with connected compact analytic fibers (Eyssidieux et al. 2012, Introduction, p. 1546). If there are no positive-dimensional compact analytic subvarieties, all those fibers are points and the reduction is an isomorphism. Theorem 1 therefore also disproves the holomorphic-convexity conjecture.

An important development was the construction of almost-holomorphic reduction maps by Campana and Kollár (Campana 1994; Kollár 1993, 1995). They describe, through a very general point, the normalized subvarieties whose fundamental groups have finite image in the ambient group. Their existence provides an algebraic and geometric description of the part of the variety that such subvarieties fill. Holomorphic convexity asks in addition for a proper holomorphic reduction of the universal cover to a Stein space. The large case makes this additional analytic requirement especially visible: the expected reduction has no positive-dimensional fibers.

Linearity of the fundamental group supports strong affirmative results. Katzarkov and Ramachandran proved holomorphic convexity for projective surfaces whose fundamental group admits a faithful complex representation with reductive Zariski closure (Katzarkov and Ramachandran 1998). Eyssidieux treated the reductive linear case in arbitrary dimension (Eyssidieux 2004). Eyssidieux, Katzarkov, Pantev, and Ramachandran proved the linear Shafarevich conjecture for smooth projective varieties (Eyssidieux et al. 2012). More recently, Bakker, Brunebarbe, and Tsimerman developed linear Shafarevich theory for normal algebraic spaces and quasiprojective varieties (Bakker et al. 2024). The construction here concerns the unrestricted fundamental group. Its infinite-order argument uses compatible local covers of an arrangement, so it does not require a faithful linear representation of that group.

A closely related line of inquiry appears in Bogomolov and Katzarkov’s constructions of projective surfaces and their potential counterexamples to the Shafarevich conjecture (Bogomolov and Katzarkov 1998, sec. 4). Their proposed obstruction uses infinite chains of compact curves and depends on group-infiniteness statements left conjectural there. Here the obstruction is a rank-three lattice cover of a simple abelian surface. The group-theoretic task is to prove that its cyclic image survives all global relations; the local path criterion supplies that infiniteness argument.

Several established tools make the construction possible. Arithmetic ball quotients supply arbitrarily large regions with exactly prescribed finite hyperplane configurations. Stover and Toledo’s virtual ramified-cover theorem supplies the required sign covers (Stover and Toledo 2022; Llosa Isenrich and Py 2025), and Zarhin’s theorem supplies a genus-two curve with simple Jacobian (Zarhin 2000). Standard hyperbolic geometry (Bridson and Haefliger 1999) underlies the local path criterion. Hamm–Lê’s quasi-projective Lefschetz theorem (Hamm and Lê Dũng Tráng 1985), together with a deformation argument, allows us to retain a prescribed surface while passing from an orbifold to a smooth projective ambient variety. Each use is stated with its hypotheses at the relevant point below.

The central construction

The principal task is to embed a simple abelian surface in a smooth projective variety so that its fundamental group has infinite cyclic image. Once such an embedding is available, an ample complete intersection in a product with a sufficiently large abelian variety gives Theorem 1. The complete intersection contains the prescribed surface, and its other fibers over the added abelian variety are finite. This separates the two reasons that a subvariety has infinite fundamental-group image: it either moves in the added abelian variety or lies in the simple surface. Section 2 proves this reduction first.

To obtain the cyclic image, we begin with a genus-two curve \(C\) whose Jacobian is simple. Its presentation as a double cover of \(\mathbb P^1\) uses four branch points in one small disk and two outside that disk. We call the corresponding meridians red and blue. A complex hyperbolic arrangement realizes this marked line as an exceptional fiber. Extra hyperplanes meeting the red part of the pencil impose relations that identify all four red meridians. The sphere relation then identifies the two blue meridians. The image of the genus-two surface group is therefore cyclic: it lies in the cyclic subgroup generated by the product of the resulting red and blue involutions.

These relations alone give only an upper bound. The central difficulty is to prove that the cyclic image is infinite after all global identifications and fillings. We use covers defined separately on large local models of the arrangement. We model their monodromies using the infinite dihedral group generated by two involutions \(r,b\); its even words \((rb)^n\) carry the integer translation label \(n\). These local monodromies need not define a representation of the full projective fundamental group. Instead, they agree on the endpoint tests needed to read short paths in overlapping charts. The graphs of these local covers are uniformly Gromov hyperbolic. A local path criterion then prevents a loop detected by one chart from becoming trivial through global relations.

The edges of these graphs are paths with bounded projected diameter, rather than bounded length. Consequently every power of a fixed loop near the distinguished fiber is a single edge. The local criterion detects every nonzero power separately and gives the infinite-order lower bound. This feature is essential to the argument; neither a global coloring nor a family of finite quotients is needed for that lower bound.

Two copies of the pencil give a distinguished fiber \(C\times C\). An order-four symmetry exchanges the factors and rotates their normal directions. Resolving its coarse quotient over this fiber produces a map from a point blowup of \(\mathop{\mathrm{Sym}}^2 C\), itself the blowup of \(\mathop{\mathrm{Jac}}(C)\) at a point. The same local argument detects the sum of the two translation labels. We then realize the resulting surface map in an ordinary smooth projective ambient variety and remove its point blowups by projective modifications. Throughout these operations we preserve the actual homomorphism from the surface group.

The local path criterion and the ambient blowdown construction are formulated separately from the example. The former converts bounded chart compatibility and hyperbolicity into nontriviality of loops. The latter realizes a point blowdown of an embedded surface inside a new smooth projective ambient variety, preserving the ambient fundamental group and the induced surface homomorphism. These two statements isolate the group-theoretic and projective-geometric mechanisms of the construction.

Structure of the proof

Section 3 proves the local path criterion independently of the geometric application. Its hypotheses are bounded-range path readings, compatibility of endpoint equality, and uniform hyperbolicity of the reading graphs; no local compactness is required. Section 4 constructs the arithmetic arrangement, its sign covers, and the distinguished surface map. Section 5 defines the local monodromy labels, proves their buffered compatibility, and establishes hyperbolicity of the associated graphs. Section 6 verifies the path criterion for the constructed orbifold and proves that the surface image is infinite cyclic. Section 7 carries this image into an ordinary smooth projective variety, removes the point blowups, and completes the proof using Section 2.

Conventions

All varieties are over \(\mathbb C\) unless a field is specified, and all fundamental groups are taken in the usual topology. For a smooth complex orbifold, equivalently a smooth Deligne–Mumford stack in the constructions below, the fundamental group means that of its classifying space. An ordinary point has trivial stabilizer. Fundamental-group image statements are understood up to a common choice of basepoints and connecting paths. Projective bundles parametrize lines. Complex hyperbolic distance is normalized so that a point of Euclidean radius \(\tanh r\) in the unit ball has distance \(r\) from its center.

Paths and their readings

All graphs have oriented edges with specified reverses; multiple edges and loops are allowed. Each edge has length one. Paths are finite edge paths, and their lengths count edges. A constant path has length zero. The path groupoid of a graph is obtained by cancelling consecutive reverse edges. We use the path metric on each connected component of a graph.

Fix a positive integer \(N\). At each vertex \(v\) of a graph \(G\), suppose that we are given a pointed graph \((D_v,d_v)\) and a rule for reading every path of length at most \(N\) starting at \(v\) as an edge path in \(D_v\) starting at \(d_v\). Write \([p]_v\) for the terminal vertex of the reading of \(p\). The following are the required properties.

1. Readings respect prefixes and reverses of edges, and read a backtrack as a backtrack. If \(p\) starts at \(v\) and \(|p|<N\), reading gives a bijection between the edges leaving the actual endpoint of \(p\) and the edges leaving \([p]_v\). If \([p]_v=[p']_v\), then \(p\) and \(p'\) have the same actual endpoint in \(G\), and their edge bijections agree whenever both are defined.

2. Suppose that \(p\) runs from \(v\) to \(w\) and that \(a,b\) start at \(w\). If each of \(p,a,b\) has length at most \(N/2\), then \[ [pa]_v=[pb]_v\quad\Longleftrightarrow\quad [a]_w=[b]_w. \tag{2}\]

3. All connected components of all the graphs \(D_v\) are Gromov hyperbolic with a common constant.

These rules concern paths within the stated range. They do not assert that the whole graph \(G\) maps to \(D_v\), or that the whole graph \(D_v\) maps to \(G\).

A loop \(p\) at \(v\) closes on reading if \([p]_v=d_v\). Let \(\mathcal Q\) be the quotient of the path groupoid of \(G\) by the relations that every such loop of length at most three is the constant path.

The proof constructs a locally geodesic representative for each path, unique up to a uniformly narrow comparison. Such representatives can be continued one edge at a time. The comparison class is unchanged by the relations defining \(\mathcal Q\), but distinguishes the edge in the lemma from the constant path.

Calculations within one chart

We first record precisely how the reading rules allow short calculations to move between charts. By (P1), any edge path in \(D_v\) starting at a state already reached by reading can be lifted to a path in \(G\), as long as the total reading length is at most \(N\). If two such readings have the same endpoint, their lifted paths have the same actual endpoint, and any further common continuation reads identically. Together with (P2), this has three useful consequences.

First, a short circuit closes on reading if and only if its two complementary paths from one corner to another have equal read endpoints. Indeed one may read one path and the reverse of the other, using (P1). Closure can be tested at any corner: a cyclic change of starting corner amounts to prefixing by a side, applying (P2), and cancelling that side using (P1). Here and below, “short” means that all prefixes and continuations in the comparison lie within the ranges of (P1) and (P2).

Second, geodesicity of a short reading is independent of a short prefix. For example, if the reading of \(a\) from \(w\) is geodesic but its reading after \(p:v\to w\) has a shorter competitor, lift that competitor from \([p]_v\). It gives a path \(b\) from \(w\) with \([pa]_v=[pb]_v\). Equation (2) gives \([a]_w=[b]_w\), contradicting geodesicity. The reverse implication uses a shorter competitor in \(D_w\) and the same argument. For distance comparison, suppose the states are reached by paths \(a,b\) from \(w\). Lift a shortest connector \(c\) between their read states, starting after \(a\) (or after \(pa\)). Its length is at most \(|a|+|b|\), and apply (P2) to \(ac\) and \(b\). Doing this in both charts proves equality of the distances whenever \(|p|\le N/2\) and \(2|a|+|b|\le N/2\). These bounds, as well as their versions with \(a,b\) interchanged, hold in every use below.

Third, a short strip of closed circuits can be calculated in one chart without adding the lengths of all its transverse paths. Here is the explicit bookkeeping. Let \(a_j,b_j\) be the prefixes down the two sides of the strip, and let \(\rho_j\) be its transverse path from the endpoint of \(a_j\) to that of \(b_j\). Starting in the chart at the beginning of the first side, the equalities to prove are \[ [a_j\rho_j]=[\rho_0b_j]. \tag{3}\] Closure of the next elementary circuit, transported after \(a_j\) by (P2), compares its two routes across the next step. Explicitly, write \(a_{j+1}=a_js_j\) and \(b_{j+1}=b_jt_j\), where \(s_j,t_j\) are the next side portions. Closure and then the preceding equality give \[\begin{aligned} [a_{j+1}\rho_{j+1}]&=[a_j\rho_jt_j]\\ &=[\rho_0b_jt_j]=[\rho_0b_{j+1}], \end{aligned}\] using (P1) for the common continuation, edge by edge. This proves (3) at the next step. Thus only the side prefixes and each individual transverse path enter the range bound. Conversely, transverse paths found in this one chart lift to \(G\) and give circuits that close when read from their own corners. These facts will justify all uses of diagrams below.

Choose an integer \(\Delta\ge10\) large enough for the following standard consequences of the common hyperbolicity bound. In every component of every chart, geodesic triangles are \(\Delta\)-slim on vertices: every vertex on one side lies within \(\Delta\) of a vertex on one of the other two sides. Dividing a geodesic quadrangle by a diagonal then shows that each side lies within \(2\Delta\) of the union of the other three sides. The Gromov product \[(x\mid y)_u=\tfrac12\bigl(d(u,x)+d(u,y)-d(x,y)\bigr)\] satisfies \[ (x\mid z)_u\ge \min\{(x\mid y)_u,(y\mid z)_u\}-\Delta. \tag{4}\] Moreover, for a geodesic segment \([x,z]\), there is a vertex on that segment at distance at most \((x\mid z)_u+10\Delta\) from \(u\). Enlarging \(\Delta\) absorbs all rounding to vertices. These are the usual equivalent forms of hyperbolicity; see (Bridson and Haefliger 1999, III.H).

We will also use the following elementary comparison. If two geodesic segments \(\gamma_1,\gamma_2\), of lengths \(\ell_1,\ell_2\), have corresponding endpoints at distance at most \(h\), then for every integer \(t\ge0\), \[ d\bigl(\gamma_1(\min\{t,\ell_1\}), \gamma_2(\min\{t,\ell_2\})\bigr)\le10(h+\Delta). \tag{5}\] This is a synchronous comparison of width \(10(h+\Delta)\): the parameters advance by one on both segments until the shorter segment ends, and then that segment waits at its endpoint. To check the bound, join the corresponding endpoints by geodesics of length at most \(h\). The resulting quadrangle is \(2\Delta\)-slim. A point close to the opposite geodesic has a mate whose distance from that geodesic’s initial endpoint differs from its own parameter by at most \(h+2\Delta\). A point close to an end connector is within \(h+2\Delta\) of the corresponding end. These estimates, together with the bound \(2h\) on the difference of the lengths, imply (5), also when one segment has ended.

Fix integers \[ H=1000\Delta,\qquad L\ge10^6H,\qquad k=20L, \qquad N\ge10^5k. \tag{6}\] Taking \(L=10^6H\), for example, gives a threshold \(N_0=10^5(20L)\) depending only on the original hyperbolicity constant. A guide is a path each of whose subpaths of length at most \(k\) reads as a geodesic in the chart at its initial vertex. Its entire length may be arbitrarily large. Every individual chart calculation below uses paths, including prefixes, of length less than \(1000k\). Consequently (P1) and (P2) always apply. A long guide is handled in separate short pieces, never by reading its whole prefix.

Narrowing comparisons between guides

For paths \(\alpha,\beta\) with the same initial and terminal vertices, a ladder of width \(h\) consists of the following data. A finite schedule of pairs of positions begins at \((0,0)\) and ends at \((|\alpha|,|\beta|)\); at each step each position advances by zero or one. At each scheduled pair there is a path, called a rung, from the vertex on \(\alpha\) to the vertex on \(\beta\), of length at most \(h\). The first and last rungs are empty. Each elementary circuit, formed by consecutive rungs and the intervening portions of the two paths, must close on reading. Pausing both paths is allowed. In particular, this definition does not require an elementary circuit to have length at most three. Ladders are comparisons of readings, rather than expressions in the relations defining \(\mathcal Q\).

For the widths used below, ladders are symmetric, by changing corners of their elementary circuits. They compose with addition of widths whenever the sum is at most \(1000H\), which suffices for every composition below. Indeed, given ladders from \(\alpha\) to \(\beta\) and from \(\beta\) to \(\gamma\), synchronize the advances along \(\beta\), inserting pauses in the other sides as necessary, and concatenate the two rungs. Each resulting circuit is a union of one or two elementary circuits from the given ladders, so closes by (3). Appending the same trailing path to both sides preserves the width: use empty rungs along that trailing path.

In particular, existence of a width-\(H\) ladder is an equivalence relation on guides with fixed endpoints. Reflexivity uses empty rungs, symmetry was noted above, and transitivity follows by composing two ladders and applying Lemma 6.

Continuing a guide by one edge

We next show that this comparison class can be continued along any edge. The difficulty is that appending an edge need not preserve local geodesicity. We repair the guide by choosing nearby vertices at widely spaced marks and minimizing the total distance between those choices. All nearby choices are specified by paths in the charts; distance in \(G\) alone would forget the states that must be compared.

The invariant of a path

We can now finish the local criterion. Fix an initial vertex \(v\) and consider guides starting at \(v\), modulo width-\(H\) ladders. By Lemma 7, appending an edge \(e\) defines a continuation of such a class. It is independent of both choices involved. If \(\alpha,\alpha'\) represent the same class, append \(e\) to their width-\(H\) ladder. If \(\beta,\beta'\) are any two guides supplied by Lemma 7, composition gives a ladder between them of width at most \[50H+H+50H=101H.\] Lemma 6 reduces this to width \(H\). The same argument covers different choices of the extension for a fixed guide.

Let \(q\) be a loop of length at most three that closes on reading. It has a ladder of width at most three with the constant path. One explicit schedule traverses \(q\) on the first side while the second waits: after a noninitial position use the remaining suffix of \(q\) as the rung, and use empty first and last rungs. The first circuit closes because \(q\) does, and the subsequent ones cancel consecutive reverse edges. The short-chart rules justify the same assertion at each corner. Appending this ladder after any guide \(\alpha\) gives a width-three ladder between \(\alpha q\) and \(\alpha\).

Successively applying Lemma 7 along \(q\) produces a guide \(\beta\) with a ladder to \(\alpha q\) of width at most \(150H\). To see the bound, at each step append the remaining common trailing edges to the comparison and compose; each of at most three steps adds \(50H\). Composing with the preceding width-three ladder and narrowing shows that \(\beta\) and \(\alpha\) represent the same class. The same reasoning applies to a backtrack, which closes on reading by (P1).

Starting from the empty guide at \(v\) and continuing along a path thus gives a class unchanged by every defining relation of \(\mathcal Q\). Insertion of a relation in the middle of a path causes no problem: the classes agree after the inserted loop, and the well-defined continuations along the remaining edges preserve that agreement.

Two pencils and their coupling hyperplanes

The two pencils will use the same six marked directions on \(\mathbb P^1\). We first choose these markings so that their double cover has simple Jacobian, and then choose the arithmetic ball containing the pencils. Choose six distinct real algebraic numbers \(\lambda_1,\ldots,\lambda_6\) with \[ |\lambda_i|<0.01\quad(1\le i\le4), \qquad \lambda_5<-4,\quad \lambda_6>4, \tag{12}\] and such that the smooth double cover \[ \pi:C\longrightarrow\mathbb P^1, \qquad y^2=\prod_{i=1}^6(t-\lambda_i), \tag{13}\] has simple Jacobian. We call the first four slopes red and the last two blue. These requirements can be imposed simultaneously, as follows.

Start with six disjoint real open intervals satisfying (12). Monic real polynomials with one root in each interval form a nonempty open set in coefficient space. At three sufficiently large distinct primes prescribe squarefree degree-six reductions with factor degrees \[(6),\qquad(5,1),\qquad(2,1,1,1,1).\] Weak approximation produces a monic polynomial in \(\mathbb Q[t]\), integral at these primes, with these reductions and with its real coefficients in the chosen open set. Its Galois group contains a six-cycle, a five-cycle, and a transposition, by the Frobenius cycle criterion. The six-cycle makes the group transitive. The stabilizer of the fixed point of the five-cycle is transitive on the remaining five points, so the group is two-transitive. Conjugating its transposition then gives every transposition, and the Galois group is \(S_6\). Let \(F\) be its splitting field. Then \(F\) is totally real, \(F\ne\mathbb Q\), and all six slopes belong to \(F\). Zarhin’s theorem gives \(\mathop{\mathrm{End}}(\mathop{\mathrm{Jac}}(C))=\mathbb Z\) over an algebraic closure, hence \(\mathop{\mathrm{Jac}}(C)\) is simple (Zarhin 2000, Theorem 2.1).

Use the inclusion \(F\subset\mathbb R\) containing the chosen roots as the distinguished real embedding. Choose a CM extension \(E/F\) and an element \(a\in F\) which is positive at this embedding and negative at every other real embedding; weak approximation supplies \(a\). On \(E^5\) put \[ h=\operatorname{diag}(1,1,1,1,-a). \tag{14}\] At the distinguished embedding the negative lines form complex hyperbolic space of dimension four. In the affine chart with last homogeneous coordinate one, write this ball as \[\mathbb B=\{(z,w)\in\mathbb C^2\times\mathbb C^2:|z|^2+|w|^2<a\}, \qquad o=(0,0).\] The metric is normalized so that in the unit-radius ball a point at distance \(r\) from the origin has Euclidean radius \(\tanh r\).

Choose \(c\in F\) with \[ 0.85<c/\sqrt a<0.9. \tag{15}\] The \(z\)-family consists of the six slope hyperplanes and two coupling hyperplanes \[H^z_i=\{z_2=\lambda_i z_1\}\cap\mathbb B\quad(1\le i\le6), \qquad K^z_\pm=\{z_1=\pm c\}\cap\mathbb B.\] Define the \(w\)-family by the same equations in \(w\). Let \(\mathcal D\) denote these sixteen labeled complex hyperbolic hyperplanes. They are all defined over \(E\) and are preserved, with their labels permuted, by \[ \sigma(z,w)=(w,-z). \tag{16}\] This is an order-four isometry represented by an integral unitary matrix. The two pencils have centers \[L_z=\{z=0\}\cap\mathbb B, \qquad L_w=\{w=0\}\cap\mathbb B.\]

The intersection pattern is elementary but important. The \(z\) slopes all meet along \(L_z\). On \(K^z_\pm\), a slope of value \(\lambda\) meets the ball exactly when \(c^2(1+|\lambda|^2)<a\). Thus each coupling hyperplane meets the four red slopes, in distinct subspaces disjoint from \(L_z\), and meets neither blue slope. The two \(z\) coupling hyperplanes are disjoint. These statements hold also in the \(w\)-family. A \(z\) coupling hyperplane and a \(w\) coupling hyperplane are disjoint because \(2c^2>a\). Every intersection involving the two families has a local product description in the \(z\) and \(w\) variables. In particular, the phrase “product” here concerns local analytic coordinates, rather than a product decomposition of the hyperbolic metric.

Figure 2 shows the incidences in one family; the second family is obtained by using the other coordinate pair.

Exact models in congruence quotients

The arithmetic lattice theorem of Borel–Harish-Chandra (Borel and Harish-Chandra 1962, Corollary 12.4), applied to (14), gives a cocompact arithmetic lattice of simplest type in \(\mathrm{PU}(4,1)\). Indeed all the other archimedean factors are compact, and the form is anisotropic over \(E\): a nonzero \(E\)-rational isotropic vector would remain isotropic at a definite conjugate embedding. Torsion-free principal congruence subgroups therefore give compact complex hyperbolic manifolds, which are projective by the Baily–Borel theorem (Baily and Borel 1966, Theorem 10.11). One can also obtain projectivity from the positive canonical bundle of a compact ball quotient. We use principal congruence subgroups normalized by \(\sigma\), and may take their levels arbitrarily deep. These standard arithmetic facts are recalled in (Stover and Toledo 2022, sec. 3.1).

Fix positive normal vectors \(n_i\in E^5\) for the members of \(\mathcal D\), scaled to have integral coordinates. We always label a translate of the \(i\)th hyperplane by \(i\), and use the prescribed normal \(\gamma n_i\) for its translate by \(\gamma\). Congruence subgroups may be chosen to avoid the finite scalar kernel of the projective action.

We next record the consequences for the divisors and centers that we will blow up. In \(M_0=\Gamma_0\backslash\mathbb B\), each label gives a closed embedded smooth totally geodesic divisor, possibly disconnected. Its lifts are locally finite and have disjoint sheets by Proposition 8. The divisors are algebraic because \(M_0\) is projective.

Within one family the intersection centers have complex codimension two. A center is either a common intersection of at least two slopes, or an intersection of a red slope with a coupling hyperplane. At a slope center only a subset of the six slopes may occur. The subset is constant along a connected center: if another slope meets it, the local model shows that it contains the intersection, and the containment extends along its lifts. Distinct centers within one family are disjoint; otherwise all participating hyperplanes would be visible near a point of intersection, contradicting the prototype pattern. Coincident centers are counted once. Thus these centers are closed embedded smooth subvarieties. Centers from different families have the product structure already described. These assertions apply to the full pulled-back arrangement after any finite cover.

Two global sign covers

For the later local models we need global parity characters on the arrangement complement. We obtain these from the following theorem of Stover–Toledo, in the form stated by Llosa Isenrich–Py (Llosa Isenrich and Py 2025, Theorem 4.2); see also (Stover and Toledo 2022).

Apply Theorem 9 with \(d=2\) to each of the sixteen label divisors separately, on the initial congruence quotient \(M_0\). Each application satisfies the disjoint-component hypothesis. We do not apply the theorem to the union of the intersecting divisors. Take a common finite unramified cover of the resulting base covers. The sum, in \(\mathbb Z/2\), of the eight characters for the \(z\) labels gives a character \(\chi_z\) on the complement of the \(z\) arrangement. It has value one on a meridian of every \(z\) hyperplane. Pull it back to the complement of both families. The same procedure gives a \(w\) character. Subsequent finite covers need not be congruence: every use of Theorem 9 has already taken place on \(M_0\).

We arrange equivariance before fixing the signs. First intersect the finitely many \(\sigma\)-conjugates of the subgroup defining our common cover, so that \(\sigma\) acts on it. Since \(\sigma^2\) preserves the \(z\)-family, the character \[\delta=\chi_z+\sigma^{2*}\chi_z\] is zero on all meridians. The kernel of the map from the fundamental group of a divisor complement to that of the ambient smooth variety is normally generated by meridians. Hence \(\delta\) factors through the fundamental group of the compact ball quotient. Pass to a further finite cover by intersecting the kernel of this character with its \(\sigma\)-conjugates. On that cover the pulled-back \(z\) character satisfies \(\sigma^{2*}\chi_z=\chi_z\). Define \[ \chi_w=\sigma^*\chi_z. \tag{18}\] Then \(\sigma\) exchanges the two characters. They have the prescribed meridian values in their respective families, and each is defined using only that family.

Write \(M=\Gamma\backslash\mathbb B\) for this final finite cover, still with the full pulled-back arrangement. The image of \(o\) in \(M\), also denoted \(o\), is fixed by \(\sigma\). It is an isolated component even of the fixed locus of \(\sigma^2\), because the derivative of \(\sigma^2\) at \(o\) is \(-\mathop{\mathrm{id}}\). No assertion of this kind is needed for the other fixed loci of the symmetry on \(M\).

Filling the arrangement

Blow up all intersection centers within the \(z\)-family, and then the transforms of those within the \(w\)-family. Product coordinates show that the operations for the two families commute. The branch rule for the \(z\) character is as follows. Every strict transform of a \(z\) hyperplane is a branch divisor of order two. If \(k\) such hyperplanes pass through a center, its exceptional meridian is the product of their meridians, and therefore has sign \(k\bmod2\). Accordingly the exceptional divisor is branched precisely when \(k\) is odd. In that case blow up each meeting of this exceptional divisor with a strict transform of a hyperplane. The two meeting divisors both have sign one; their new exceptional divisor has sign zero. The branch components within this family are now smooth and disjoint. Apply the identical rule to the \(w\) character. This prescription includes partial pencils. In particular an ordinary red–coupling pair has \(k=2\), so its first exceptional divisor is unbranched.

Let \(\widehat M\) denote the resulting smooth projective base. Extend the two sign covers from the complement by normalization over \(\widehat M\), and take them together. Finite topological covers of smooth complex algebraic varieties algebraize, and normalization in the resulting finite extensions gives finite algebraic covers; thus this procedure is projective. Locally along a branch divisor the normalization is the smooth double-cover model \(t=u^2\). Branch divisors within a family are disjoint, and across the families the local equations and the two signs are independent. Their combined normalization is therefore smooth, including at crossings, where it has product equations \((t_1,t_2)=(u_1^2,u_2^2)\). We obtain a smooth projective variety \[ V\longrightarrow\widehat M \tag{19}\] with deck group \((\mathbb Z/2)^2\). It is connected: a meridian at a general point of a \(z\) divisor has sign \((1,0)\), and a meridian at a general point of a \(w\) divisor has sign \((0,1)\), so the combined monodromy is surjective.

Near \(o\) all six slopes in each pencil occur and the coupling hyperplanes are absent after the neighborhood has been made small. The blowups are the product of the blowups of the two coordinate planes at their origins. The exceptional fiber of \(\widehat M\to M\) is \(\mathbb P^1\times\mathbb P^1\). Each exceptional line has six branch markings and has even meridian sign. Its inverse image in \(V\) is consequently \[ F_*=C\times C. \tag{20}\] Put \[L=\pi^*\mathcal O_{\mathbb P^1}(-1), \qquad N=N_{F_*/V} =\operatorname{pr}_1^*L\oplus\operatorname{pr}_2^*L.\] In fact a neighborhood of \(F_*\) has the product line-bundle description provided by these two tautological lines. Indeed, in each factor, after the unbranched radial disks have been filled, their disk bundle retracts onto its zero section. Away from the six marked directions its cover is thus pulled back from the direction line. Normalization gives the six-branch cover in that direction variable, with the radial line bundle pulled back unchanged.

The equivariance (18) makes the kernel of the combined sign character invariant, so \(\sigma\) lifts on the complement. The equivariant blowups and normalization extend the lift to an automorphism \(s\) of \(V\). Its restriction to \(F_*\) exchanges the two copies of \(C\), possibly followed by either hyperelliptic deck involution. Composing with a deck transformation makes this restriction the pure swap. To determine the lift in the whole local model, note that \(u,v\) are actual tautological vectors: the covering map in these coordinates is \[(p,q;u,v)\longmapsto(\pi(p),\pi(q);u,v).\] The pure swap determines the lifted direction maps on the connected local cover. Since the base map sends the radial vectors to \((v,-u)\), there is no further radial multiplier. Thus \[ s(p,q;u,v)=(q,p;v,-u). \tag{21}\] Its fourth power is a deck transformation fixing \(F_*\) pointwise, so \(s^4=1\); a nonidentity element of \((\mathbb Z/2)^2\) does not fix \(C\times C\) pointwise. Its square is the identity on \(F_*\) and is multiplication by \(-1\) on both normal lines.

An ordinary neighborhood of the distinguished fiber

The quotient stack \([V/\langle s\rangle]\) is a smooth proper orbifold with projective coarse space, but a surface mapping into its distinguished fiber would meet stabilizers. We now replace just that fiber by an ordinary smooth resolution. Retaining the quotient stack elsewhere allows all other fixed loci to be left in place.

First blow up \(F_*\) in \(V\), and take the coarse quotient by \(\langle s^2\rangle\). The exceptional divisor is \[ P=\mathbb P_{F_*}(N), \tag{22}\] where projectivization parametrizes lines. Before taking the quotient, \(s^2\) acts trivially on \(P\) and as \(t\mapsto-t\) on its transverse coordinate. The quotient is thus smooth near \(P\), with transverse coordinate \(t^2\).

The residual involution induced by \(s\) has fixed points near \(P\) only in \(P\): away from the fiber, a fixed point would project to a nontrivial fixed point of the symmetry near \(o\) in \(M\). Its fixed points in \(P\) project to the diagonal of \(C\times C\). On the projective normal line over that diagonal, it acts by \[[u:v]\longmapsto[v:-u].\] There are exactly two eigensections, with eigenvalues \(i\) and \(-i\) before projectivization. These give two disjoint smooth fixed curves. On the three normal directions to either curve the involution acts by \(-1\): one direction is antisymmetric in the base, one is tangent to the projective fiber, and the transverse coordinate to \(P\) has eigenvalue \((\pm i)^2=-1\). Blow up both fixed curves and take the coarse quotient by the residual involution. This quotient is smooth near the distinguished fiber. Indeed, for the linear local model \[(x,t_1,t_2,t_3)\longmapsto(x,-t_1,-t_2,-t_3),\] blowing up the fixed curve \(\{t_1=t_2=t_3=0\}\) makes the action a reflection in the coordinate normal to the exceptional divisor, and the coarse quotient is smooth. Finite-order holomorphic actions are locally linearizable, so this model applies.

All centers just used are closed and lie over \(o\). The blowups and finite coarse quotients can therefore be performed globally, producing a projective coarse space \(S_0\) which modifies \(V/\langle s\rangle\) only over \(o\). The intermediate coarse spaces may retain quotient singularities elsewhere; the preceding calculation establishes smoothness on a neighborhood of the distinguished fiber. To form \(\mathcal S\), use this smooth scheme near that fiber and use \([V/\langle s\rangle]\) away from it. This is an algebraic gluing. More explicitly, in \(M\) remove all components of the nontrivial fixed loci other than the isolated point \(o\), and take the resulting invariant Zariski neighborhood. Outside \(o\) in this neighborhood the action is free, so both constructions restrict to the same ordinary quotient. Their gluing is a smooth connected effective orbifold \(\mathcal S\) with coarse space \(S_0\). The map \(\mathcal S\to S_0\) is separated and proper: these properties are local on \(S_0\), where it is either the coarse map of a finite-group quotient stack or the identity. Since \(S_0\) is projective, \(\mathcal S\) is separated and proper over \(\mathbb C\). In particular the whole distinguished fiber lies in the ordinary smooth locus of \(\mathcal S\).

The surface mapping into the ordinary fiber

Let \(A=\mathop{\mathrm{Jac}}(C)\). For a genus-two curve, the Abel map \[\mathop{\mathrm{Sym}}^2C\longrightarrow\mathop{\mathrm{Pic}}^2(C)\simeq A\] is the blowup at the canonical class, after choosing the displayed translation. To recall the geometry, Riemann–Roch gives a unique effective divisor in every degree-two class except \(K_C\); the latter has the pencil \(E_C=|K_C|\simeq\mathbb P^1\). For the degree-two quotient \(q:C\times C\to\mathop{\mathrm{Sym}}^2 C\), the inverse image of \(E_C\) is the graph \(\Gamma_\iota\) of the hyperelliptic involution. The map \(q\) is generically unramified along this graph, so \(q^*E_C=\Gamma_\iota\) as divisors; ramification at its six diagonal points does not change the generic multiplicity. The projection formula gives \[2E_C^2=(q^*E_C)^2=\Gamma_\iota^2=2-2g(C)=-2.\] The Abel map is therefore a birational morphism between smooth surfaces with one exceptional \((-1)\)-curve, and is the stated point blowup.

Choose a nonzero rational section \(\tau\) of \(L\). On the open subset of \(C\times C\) where both values are defined and nonzero, put \[ (p,q)\longmapsto[\tau(p),i\tau(q)]\in P. \tag{23}\] This section is equivariant for swapping the two factors and the involution on \(P\). Indeed (21) gives \[[\tau(p),i\tau(q)]\longmapsto[i\tau(q),-\tau(p)] =[\tau(q),i\tau(p)].\] Away from the diagonal it avoids the two fixed curves, and thus gives a rational map from \(\mathop{\mathrm{Sym}}^2C\) to the smooth resolution constructed above. All the bundles and maps in this description are algebraic: the normal-bundle identification is algebraic as well as analytic, by GAGA on the projective fiber. Resolve the rational map by a finite sequence of point blowups of its smooth projective source, using the projective coarse space as target. The resolved image remains in its closed distinguished fiber, where the coarse space agrees with the ordinary locus of \(\mathcal S\). We therefore obtain a morphism \[ f:Y\longrightarrow\mathcal S \tag{24}\] whose image lies entirely over \(o\). Thus \(Y\) is an iterated point blowup of the simple abelian surface \(A\), and \(f(Y)\) lies in the ordinary locus.

All parameters used in the finite arrangement have now been fixed. The depth of \(\Gamma_0\) has not: Proposition 8 permits its choice after the constants required for the finite-model graphs in the next section. Passing to the finite covers which produce the signs preserves these estimates. This order of choices will allow the local calculations to establish \[\mathop{\mathrm{im}}(f_*:\pi_1(Y)\to\pi_1(\mathcal S))\simeq\mathbb Z.\]

Dihedral labels on a finite arrangement

For a subset \(T\subseteq\mathcal D\), set \[U_T=\mathbb B\setminus\bigcup_{H\in T}H.\] There are characters \(\epsilon_z,\epsilon_w:\pi_1(U_T)\to\mathbb Z/2\), with \(\epsilon_z\) equal to one on a meridian of a \(z\) hyperplane and zero on a meridian of a \(w\) hyperplane, and conversely for \(\epsilon_w\). One can define each character by the parity of the winding number of the product of defining affine equations of the hyperplanes in its family. Let \(E_T\to U_T\) denote the associated four-sheet cover; it may be disconnected. These signs restrict canonically to any ball patch, up to a choice of sheets. Indeed, meridians normally generate the complement group in a simply connected ball, as is seen by filling a loop with a disk transverse to the hypersurfaces. Thus their values determine a sign character there.

We refine these signs using the infinite dihedral group \[D_\infty=\langle r,b\mid r^2=b^2=1\rangle, \qquad \epsilon(r)=\epsilon(b)=1.\] Its even subgroup is the infinite cyclic translation subgroup \(\langle rb\rangle\). Conjugation in \(D_\infty\) preserves this subgroup and changes its integer coordinate at most by a sign.

We first construct \(q_z:\pi_1(U_T)\to D_\infty\). The construction ignores all \(w\) hyperplanes. If \(T\) contains fewer than all six \(z\) slope hyperplanes, define \[q_z(\gamma)=r^{\epsilon_z(\gamma)}.\] Suppose instead that the whole \(z\) pencil is present. With only its six hyperplanes removed, projection to the direction \(\lambda=z_2/z_1\) gives a map to the six-punctured projective line. Choose its standard meridians so that the four red punctures occur first, based together in a small red disk, and the two blue punctures occur last. Assign \(r\) to each red meridian and \(b\) to each blue meridian. The sphere relation is respected because \(r^4b^2=1\). We fix this choice of the direction-line representation once, and use the same choice for every complete pencil and for both families.

It remains to add the coupling hyperplanes that belong to \(T\). They lie over the red-direction disk: on \(z_1=\pm c\) in the ball, \[|\lambda|<0.7.\] Over \(|\lambda|<1.5\), the pencil local system has a reduction of structure group to \(\langle r\rangle\). In this reduction, multiply its transition functions by the two-sheet systems defined, for the coupling hyperplanes that are present, by \[ (1-z_1/c)^{1/2},\qquad (1+z_1/c)^{1/2}. \tag{25}\] This means adding their winding parities to the exponent of \(r\). Over \(|\lambda|>1\) one has \(|z_1|<c\), so both systems in (25) have the distinguished trivializations obtained by the branch of the square root near \(1\). These trivializations identify the modified system with the unmodified pencil system on \(1<|\lambda|<1.5\). They therefore glue the two systems. This constructs \(q_z\), up to conjugacy, on all of \(U_T\). Its parity is \(\epsilon_z\): this is true on every meridian, and meridians normally generate. Interchanging \(z\) and \(w\) gives \(q_w\).

On each component of \(E_T\), both representations take values in translations. Choose integer coordinates on the translation groups and write the resulting homomorphisms as \[\ell_z,\ell_w:\pi_1(E_T)\longrightarrow\mathbb Z.\] The notation is componentwise. Their individual kernels are independent of conjugating the dihedral representations or changing sheets. The signs of the integer coordinates are immaterial until we form a sum in the central model.

For each \(T\), apply the intersection-center blowups, parity branching, and normalization of Section 4.4 to the arrangement \(T\) in \(\mathbb B\) and its sign cover \(E_T\). The resulting smooth space is its filled finite model.

The sum cover at the distinguished fiber

For the full model \(T=\mathcal D\), let \(\overline E\) denote its filled sign cover. Its fiber over \(o\) is \(F_*=C\times C\), and it has the lift \(s\) of \(\sigma(z,w)=(w,-z)\) described in Section 4. On \(F_*\), the action of \(s\) interchanges the factors. The first translation label is nonzero on the first factor and zero on the second; the reverse holds for the second label. For example, the product of one red and one blue meridian has even parity and label \(rb\), so its lift to \(C\) gives a nonzero translation. This calculation on \(F_*\) agrees with that on the complement: push a representative path slightly in a nonzero normal direction, keeping its directions away from the branch points. The coupling square roots are trivial there.

To pass from this filled model to the resolved distinguished fiber, we need an ordinary covering across the fixed points of the coarse quotient. The descent argument below will achieve this by making the lifted symmetry commute with deck translations. We will orient the two labels so that their sum is \(s\)-invariant.

The construction of \(q_z\) is unchanged under \(z\mapsto-z\), with the two coupling hyperplanes interchanged. Indeed this map fixes the direction \(\lambda\), interchanges the functions in (25), and preserves their distinguished trivializations. The analogous statement holds for \(w\). Consequently \(s\) interchanges the two translation homomorphisms up to signs. Orient them by the preceding comparison on \(F_*\) so that \[\ell_z\circ s_* =\ell_w, \qquad \ell_w\circ s_* =\ell_z.\] The nonzero restrictions to the two factors determine these signs. In particular the homomorphism \[ \ell=\ell_z+\ell_w:\pi_1(\overline E)\longrightarrow\mathbb Z \tag{26}\] is \(s\)-invariant.

The model using both kernels will be called an individual model. The full model using the sum kernel and then the lifted \(s\) quotient will be called the central model. The distinction is essential: the sum is used only near a distinguished fiber.

How kernels compare on bounded balls

Before constructing graph metrics, we prove that these local labels can be compared without a global coloring of the arithmetic arrangement. There are two possible sources of disagreement: normal-matching isometries may not be unique, and a larger chart may contain hyperplanes absent from a smaller one.

First suppose the prescribed normal vectors of a complete pencil have been matched as in Proposition 8. Two distinct slope normals span its positive normal plane and specify its two coordinate functionals, hence the ratio \(z_2/z_1\). Thus the direction-line local system is independent of the choice of an isometry extending the normal match. If a coupling normal is also matched, its homogeneous linear equation, together with those two functionals, specifies the ratio \(z_1/c\). Explicitly, if \(X_1\) is the first coordinate functional, \(T_0\) is the last homogeneous coordinate, and \(F_c=X_1-cT_0\) defines the positive coupling hyperplane, then \[z_1/c=\frac{X_1}{cT_0}=\frac{X_1}{X_1-F_c}.\] The negative coupling gives the same conclusion with the signs reversed. The gluing rule (25) is therefore intrinsic to these matched data as well. The corresponding assertions hold for the \(w\) family. Under \(\sigma\), the two families are interchanged and the \(w\) coordinates are negated. If the chosen representative of a relabeled normal differs by a scalar, use that same scalar on both sides of the match. The resulting coordinate ratios transform as just described, so the individual kernels are carried to the corresponding individual kernels. This is the equivariance needed below; it does not require a globally chosen orientation of a translation label.

Next, consider a coupling hyperplane absent from a metric ball \(Q\). A metric ball in the ball model is a Euclidean ellipsoid and is convex. Its \(z_1\) image is therefore convex and avoids the relevant value \(c\) or \(-c\). On that image the corresponding function in (25) has a single-valued square root. Where the image meets \(|z_1|<c\), choose its sign to agree with the distinguished square root. The intersection is convex, so one choice works throughout it. If the intersection is empty, there is no overlap requiring a prescribed sign. This trivializes the extra coupling system compatibly with the red-disk gluing. Hence including or omitting a coupling hyperplane missing \(Q\) gives isomorphic label systems on \(Q\).

Graphs defined by projected diameter

We now turn the local covers into graphs to which the path lemma applies. Over each connected component of \(E_T\), let \(\widehat E_T\) be the connected regular cover defined by \(\ker\ell_z\cap\ker\ell_w\). For the full arrangement we also use the cover defined by \(\ker(\ell_z+\ell_w)\) and its lifted-\(s\) quotient. Restrict all these covers to the open arrangement complement when defining graph vertices.

For \(u>0\), vertices are all points of the relevant covering space. An oriented edge is a parametrized continuous path whose projection to \(\mathbb B\) has diameter strictly less than \(u\); its reverse is the reverse path. Multiple edges are retained. In the central case we take the quotient graph by \(\widehat s\). The symmetry acts freely on the open arrangement complement, so edges out of a quotient vertex correspond bijectively to edges out of any chosen representative. Denote these graphs by \(\mathcal G_T(u)\), with \(\mathcal G_{\mathrm{cen}}(u)\) for the central quotient.

The use of diameter has an important consequence. A word of arbitrary length in loops supported in one fixed bounded region can be one edge. It is this feature that prevents the unbounded translation labels from producing large flat regions in the graphs.

The proof has three parts. First, paths carrying arbitrary deck translations inside a fixed bounded region give bounded graph diameter over every bounded radial region. Next, an arrangement-preserving collar identifies the remaining vertices with points of a covering of the sphere complement, together with a radial coordinate; the angular diameter permitted by a graph edge decays exponentially with that coordinate. Finally, comparison with the cone graph defined below proves uniform hyperbolicity. No properness or local finiteness of the graphs is required.

The arrangement at the sphere at infinity

Rescale the ball to unit radius and write \[x=\tanh(r)\xi,\qquad |\xi|=1.\] Let \(\Sigma\) be its unit sphere. Each affine intersection of the hyperplanes that reaches \(\Sigma\) is transverse to \(\Sigma\). Indeed its homogeneous Hermitian subspace is nondegenerate by the arithmetic normal-space property in the proof of Proposition 8. More explicitly, write an affine intersection as \(a_0+V_0\) with \(a_0\perp V_0\). On its homogeneous span, the Hermitian form is \[|v|^2+(|a_0|^2-1)|t|^2.\] Nondegeneracy excludes \(|a_0|=1\), precisely the tangency case. If the intersection reaches the sphere, then \(|a_0|<1\), and the intersection is transverse there. This also excludes intersections supported only on the boundary.

There is an arrangement-preserving product collar near \(\Sigma\). Here is a direct construction, including the estimate needed for the graph metric. At each \(\xi\in\Sigma\), choose a constant real vector tangent to all affine hyperplanes through \(\xi\) and with positive outward radial component. Transversality of their intersection supplies this vector. Use a small neighborhood missing every other hyperplane. A partition of unity combines these local vectors into a smooth vector field tangent to each hyperplane and transverse to the spheres. Normalize it so that the derivative of the Euclidean radius is one. Its flow yields, for \(r\ge r_0\) with \(r_0\) sufficiently large, diffeomorphisms between the ideal arrangement complement and the complement on the radius-\(r\) sphere. They preserve all arrangement strata. Because the vector field is bounded on a compact collar, their angular displacement from the original ideal point is \[ O(1-\tanh r)=O(e^{-2r}) \quad\hbox{in Euclidean distance.} \tag{27}\] In the central model, average the field under \(\sigma\) before normalizing. This retains tangency and positive radial component and makes the collar equivariant. The collar lifts to each covering; the lifted end is a radial product with a possibly disconnected covering \(I\) of the ideal complement.

On \(\Sigma\) use the metric \[d_K(\xi,\eta)=|1-\langle\xi,\eta\rangle|^{1/2}.\] For completeness, it satisfies the triangle inequality. If \(a=d_K(\xi,\zeta)\) and \(b=d_K(\zeta,\eta)\), then \(|\xi-\zeta|\le\sqrt2a\) and \(|\zeta-\eta|\le\sqrt2b\). Expanding the Hermitian inner product around \(\zeta\) gives \[|1-\langle\xi,\eta\rangle| \le a^2+b^2+|\xi-\zeta|\,|\eta-\zeta| \le(a+b)^2.\] Positivity and symmetry are immediate. In particular, (27) is an \(O(e^{-r})\) displacement in \(d_K\).

The ball distance in our normalization satisfies \[ \cosh d(x,y)= \frac{|1-\langle x,y\rangle|} {\sqrt{1-|x|^2}\sqrt{1-|y|^2}}. \tag{28}\] Consequently, for points of radii \(r+O(1)\), bounded hyperbolic distance is equivalent, with uniform constants, to angular \(d_K\) distance \(O(e^{-r})\). To check both implications, write \(x=\rho\xi\), \(y=\tau\eta\), where \(\rho=\tanh r\) and \(\tau=\tanh r'\). Then \[1-\langle x,y\rangle =1-\rho\tau+\rho\tau(1-\langle\xi,\eta\rangle).\] The second parenthesis has nonnegative real part. Thus the absolute value is at least \(\rho\tau d_K(\xi,\eta)^2\) and at most \(1-\rho\tau+d_K(\xi,\eta)^2\). For \(|r-r'|\) bounded and both radii large, the denominator in (28) and \(1-\rho\tau\) are comparable to \(e^{-2r}\). These inequalities prove the assertion. Conversely a bounded ball distance bounds \(|r-r'|\) by the triangle inequality for radial distance, so no separate radial hypothesis is needed in that direction.

On the lifted ideal complement \(I\), define \[ d_I(\xi',\eta')= \min\left\{1,\inf_\alpha\mathop{\mathrm{diam}}_{d_K}(\pi\alpha)\right\}, \tag{29}\] where \(\alpha\) runs over paths in \(I\) joining \(\xi'\) to \(\eta'\) and \(\pi\) is projection to \(\Sigma\). Put \(d_I=1\) for points in different components. The triangle inequality follows by concatenation, because the projected paths meet and the diameter of their union is at most the sum of their diameters. Distinct projected endpoints give positivity directly. For distinct lifts of the same endpoint, take a small evenly covered neighborhood in the ideal complement. A path changing the lift must leave this neighborhood, so its projected diameter has a positive lower bound. Therefore \(d_I\) is a metric, bounded by one.

A cone over an arbitrary bounded metric space

We isolate the metric calculation, a discrete form of the familiar hyperbolic-cone construction; compare (Bonk and Schramm 2000, sec. 7). We give the proof for arbitrary bounded metric spaces, as no compactness hypothesis is available here. For a metric space \((I,d_I)\) with \(d_I\le1\), let \(\mathcal C(I)\) have vertices \((\xi',t)\), \(\xi'\in I\), \(t\in\mathbb Z_{\ge0}\). Add vertical unit edges between successive heights at the same point, and horizontal unit edges at height \(t\) whenever \(d_I(\xi',\eta')\le e^{-t}\). In particular all vertices at height zero are pairwise adjacent.

Comparison with the model graphs

We can now complete the proof of Proposition 15. Work first before the finite symmetry quotient, in a connected component. Using the collar coordinates, send a cone vertex \((\xi',t)\) to the covering point of radius \(r_0+t\) above its ideal coordinate. A horizontal edge at positive height is represented by a path whose projected ideal diameter is at most \(2e^{-t}\): the factor two allows the infimum in (29) to be unattained. Place that path at radius \(r_0+t\) with the collar. The angular estimates and (28) give a uniform bound on its projected ball diameter. Vertical edges likewise have uniformly bounded projected diameter. Enlarge \(u\) to exceed these bounds strictly. At height zero, even when ideal components are different, Lemma 16 gives a uniform bound on the graph distance of the two images. The cone map therefore sends adjacent vertices to pairs at uniformly bounded graph distance.

In the opposite direction, round a point of radius at least \(r_0\) to the nearest integer layer, retaining its lifted ideal coordinate. Map the bounded radial part to one fixed height-zero vertex. The two maps are coarse inverses. Radial rounding has uniformly bounded graph displacement, and the bounded-core lemma handles all other points. To verify that this inverse also sends edges to pairs of bounded distance, consider an edge whose projection lies sufficiently far out. If one endpoint has radius \(r\), its whole projected path has radii in \([r-u,r+u]\). Formula (28) bounds the angular diameter of this path by \(O_u(e^{-r})\). The collar changes this by another term of the same order. Following the lifted path in the collar consequently gives \[d_I(\xi',\eta')\le C_u e^{-r}.\] Its rounded heights differ by a bounded amount, so (30) bounds their cone distance uniformly. If an edge meets the bounded radial part, both endpoints have bounded radius and the same conclusion follows through height zero. Hence the two graphs are quasi-isometric. Hyperbolicity of the cone and quasi-isometry invariance for geodesic spaces (Bridson and Haefliger 1999, III.H) prove hyperbolicity of the model graph.

For the central model, the collar and the comparison maps on the end are equivariant under \(\widehat s\). The lifted action preserves \(d_I\), since its projection is unitary and hence preserves \(d_K\). For the finite group \(H=\langle\widehat s\rangle\), the formula \[\overline d_I(H\xi',H\eta') =\min_{h\in H}d_I(\xi',h\eta')\] defines a metric on \(I/H\). Positivity uses finiteness of \(H\), and the triangle inequality follows by translating and concatenating minimizers. The quotient of the cone graph is the cone graph of this quotient metric, possibly with multiple edges and loops, which do not affect vertex distances. The preceding comparison thus descends, while the quotient radial core remains bounded. Lemma 17 proves hyperbolicity also in the central case. There are only finitely many subsets \(T\) and one central model, so one choice of \(u\) and one hyperbolicity constant work for all of them.

Finally, let an additional locally finite union of proper complex analytic subsets be omitted. Removing vertices and restricting edges cannot decrease distances. For the reverse inequality, take a finite edge path whose endpoints are retained. Perturb its finitely many junctions and constituent paths, relative to the endpoints, to miss the omitted loci. General position permits this because proper complex analytic subsets have real codimension at least two. Locally finite omissions cause no extra difficulty: a compact path meets only finitely many members, and the perturbation can be made in finitely many coordinate neighborhoods. Make this perturbation as a homotopy of the whole concatenated chain in the existing arrangement complement, relative to its two overall endpoints. Homotopy lifting preserves those two endpoints in the cover; the intermediate junctions move coherently. Each original projected path has diameter strictly smaller than \(u\), so a sufficiently small perturbation preserves that strict inequality. The perturbed path has exactly the same number of edges. Thus all distances between retained vertices are unchanged. For a central quotient edge path, lift the whole chain to the cover, perturb it relative to its overall endpoints, and project back; invariance of the omitted set ensures that the projected chain avoids it. The four-point inequality for hyperbolicity restricts to the retained vertex set with the same constant, proving the last assertion of the proposition.

We now fix such a value of \(u\), a common hyperbolicity constant, and a reading radius \(N\) supplied by Lemma 5. Only after these choices will we choose the arithmetic level and the geometric sizes of the charts.

Surface loops and the cyclic upper bound

Recall the distinguished fiber \(F_*=C\times C\) in \(V\), the order-four lift \(s\), and the rational section used to construct \(Y\) in Section 4.6. Put \[\mathcal K=[V/\langle s\rangle].\] There is a natural homomorphism \[ \pi_1(\mathcal K)\longrightarrow\pi_1(\mathcal S). \tag{31}\] Indeed, the substack \([F_*/\langle s\rangle]\) has complex codimension two in the smooth stack \(\mathcal K\). Removing it does not change the orbifold fundamental group, and its complement is the open substack on which the construction of \(\mathcal S\) leaves \(\mathcal K\) unchanged.

Choose a general point \(q_0\in C\). The map \[C\longrightarrow\mathop{\mathrm{Sym}}^2 C\longrightarrow\mathop{\mathrm{Jac}}(C), \qquad p\longmapsto p+q_0,\] is, up to translation, an Abel map. Its map on fundamental groups surjects onto \(\pi_1(\mathop{\mathrm{Jac}}(C))=H_1(C,\mathbb Z)\). Since point blowups preserve fundamental groups, loops from \(C\times\{q_0\}\) supply generators of \(\pi_1(Y)\). They may be represented by a finite bouquet avoiding the branch points, \(p=q_0\), the zeros and poles of the rational section, and the finitely many points where the rational map or its resolution must be avoided. Removing finitely many points from a smooth curve is surjective on fundamental groups, so these restrictions do not lose any generators.

We next compute an upper bound before taking the quotient by \(s\). Write \(H=(\mathbb Z/2)^2\) for the sign deck group. The map \[V\longrightarrow[V/H]\] is an orbifold covering, so it induces an injection of \(\pi_1(V)\) into \(\pi_1([V/H])\). This remains true over branch points: the target is the quotient stack, which retains their isotropy groups.

The pencil through \(L_z\) gives the orbifold line \([C/(\mathbb Z/2)]\), with six order-two points. Number the four red points first. Its fundamental group has the presentation \[ Q=\langle h_1,\ldots,h_6\mid h_i^2=1,\ h_1h_2h_3h_4h_5h_6=1\rangle. \tag{32}\] The kernel of the parity character sending every \(h_i\) to \(1\in\mathbb Z/2\) is \(\pi_1(C)\). With the \(w\) sign point fixed over a general direction, the map of this orbifold pencil to \([V/H]\) identifies the homomorphism from this even subgroup with the homomorphism from \(\pi_1(C\times\{q_0\})\) to \(\pi_1(V)\), followed by the injection just described.

This argument allows the cyclic image to be finite. We will exclude that possibility by applying Lemma 5 to paths in a dense open subset of \(\mathcal S\).

Paths in the ordinary open set

Let \(O\subset\mathcal S\) be the sign-covered arrangement complement, quotiented by \(s\), after also removing the full inverse image of the fixed loci in \(M\) of all nonidentity powers of \(\sigma\). The modifications defining \(\mathcal S\) are isomorphisms over this set. In particular \(O\) is a dense connected smooth variety, and all quotient actions used over it are free.

There is a useful space for specifying lifts. Over the complement in \(\mathbb B\) of the lifted arrangement and the lifted omitted fixed loci, pull back the sign data from \(V\), and call the resulting space \(O^\#\). Then \[O^\#\longrightarrow O\] is an ordinary covering. Changes of lift are generated by the ball deck group \(\Gamma\) and by \(\sigma\), the latter acting on the sign data through \(s\). These are precisely the transformations used to form \(O\); we do not divide out the two sign deck transformations. The group generated by \(\Gamma\) and \(\sigma\) acts freely on the retained ball region.

Fix the diameter bound \(u\) supplied by Proposition 15. Define a graph \(G\) whose vertices are the points of \(O\). An edge is a parametrized continuous path in \(O\) whose lifted projection to \(\mathbb B\) has diameter strictly less than \(u\). All graphs use the same parametrization and reversal conventions. The diameter condition is independent of the lift, because changes of lift act by ball isometries. No bound is imposed on path length or winding. In particular, arbitrarily high powers of a loop can be individual edges when their projections stay in one small region.

Choosing and comparing the charts

We make the order of the parameters explicit. First fix \(u\), then a common hyperbolicity constant for the model graphs, and then the integer \(N\) required by Lemma 5. Set \[ R=(2N+100)(u+1). \tag{33}\] A lift of a graph path with at most \(N\) edges stays within distance \(Nu\) of its initial ball point, regardless of the lengths of the individual parametrized paths.

The special points of \(\mathbb B\) are the points of \(\Gamma o\), the lifts of the distinguished point of \(M\). Choose \(J>10R\) so large that, in the central model, both of the following hold at every point of distance at least \(J-3R\) from \(o\):

1. On sign loops projected into the \(R\) ball about that point, the kernel of the sum translation equals the intersection of the two individual translation kernels.

2. For \(j=1,2,3\), the displacement by \(\sigma^j\) exceeds \(10R\).

The first requirement follows from Proposition 14. For the second, none of \(\sigma,\sigma^2,\sigma^3\) has a boundary fixed point. The ball distance formula then implies that its displacement tends uniformly to infinity as the viewpoint tends to the boundary.

Let \(B(R)\) be the buffer radius in Proposition 14, enlarged so that \(B(R)>R\), and choose \[ K>10\bigl(J+R+B(R)\bigr). \tag{34}\] All these constants depend only on the finite models. Now take the initial arithmetic level deep enough that Proposition 8 holds on balls of radius \(10K\), that such balls inject into the torsion-free ball quotient, that the full finite arrangement agrees with the lifted arrangement in these neighborhoods of every special point, and that distinct special points have distance greater than \(100K\). Enlarge this last choice, if necessary, to include the fixed compact paths used in Lemma 20. Subsequent finite covers preserve all these requirements.

For a vertex \(v\in G\), choose a lift with ball coordinate \(x\) and its actual sign data. Its chart has one of two forms.

Central chart.

If \(\mathop{\mathrm{dist}}(x,\Gamma o)<J\), use the full finite model centered at the unique nearby special point. Read in the sum-translation cover and then quotient by the lifted order-four symmetry, as in Lemma 12.

Individual chart.

Otherwise take all lifted hyperplanes meeting \(B_{\mathbb B}(x,K)\), identify them, with their prescribed normal vectors, with the corresponding subset of \(\mathcal D\), and use the model cover given by the two individual translation kernels. There is no symmetry quotient in this chart.

These chart graphs are the graphs of Proposition 15, with the appropriate additional analytic omissions. One can transport the whole locally finite union of lifted omitted fixed loci to model coordinates; any extra lifted arrangement hyperplanes outside the comparison region may be omitted as well. Such omissions are globally defined on the model ball. In central coordinates they are invariant under the symmetry. Proposition 15 shows that these omissions do not enlarge the uniform hyperbolicity constant.

For central charts, fix the identifications of sign data once on the large comparison ball about \(o\). Sign characters on that ball complement are determined by their meridians. Choose the sheet identifications to agree with the local construction near \(F_*\); they then intertwine the actual lift \(s\) with the model lift throughout this connected region. Transport the choices to all special points by \(\Gamma\). These choices are also compatible with \(\sigma\), by the symmetry of the construction.

Within the comparison ball about a special point, the only quotient transformations that can identify two ball points are the four symmetry powers centered there. Indeed, if \(g\in\langle\Gamma, \sigma\rangle\) identifies two such points, it carries the special point to another one at distance at most twice the comparison radius. Separation forces it to fix that special point. Its stabilizer is exactly the corresponding conjugate of \(\langle\sigma\rangle\), since \(\Gamma\) is torsion free.

To read a path of at most \(N\) graph edges from \(v\), lift it from the chosen point of \(O^\#\). Its ball projection remains in the comparison region. Identify the resulting sign path with the model sign path, lift further from a chosen initial point in the translation cover, and, in a central chart, take the class modulo the lifted symmetry. This is a path in a model graph \(D_v\), with initial state \(d_v\). Write \([p]_v\) for its terminal state.

Central models are centered only at points of \(\Gamma o\). Other full pencil fibers, including those over \(\Gamma_0o\setminus\Gamma o\), and the other quotient fixed loci remain unmodified in \([V/\langle s\rangle]\). An ordinary-open loop around such a locus may have a ball or sign lift with different endpoints; an individual chart retains this difference. Equality of actual endpoints is not required to imply equality of read states. For orbifold relations, the relevant meridian power lifts to a closed loop in a smooth chart in \(V\); its closure will be checked separately below.

From orbifold relations to path relations

Let \(\mathcal P(G)\) be the path groupoid of \(G\) modulo backtracking and every edge loop of at most three edges that closes in its starting chart. Lemma 5 and Lemma 21 detect nontrivial elements of this groupoid. To use that conclusion for \(\mathcal S\), we need the following direction of comparison: every loop that is trivial in \(\pi_1(\mathcal S)\) must already be trivial in \(\mathcal P(G)\).

We recall explicitly the topological fact about orbifolds used in this comparison.

Combining the last two lemmas with Lemma 5 gives \[ \text{a single edge loop with distinct read endpoints represents a nonidentity element of }\pi_1(\mathcal S). \tag{36}\] This conclusion does not require any of the translation labels to extend to all of \(\pi_1(\mathcal S)\).

Infinite order and completion of the proof

Choose a based loop \(\alpha\) in \(C\times\{q_0\}\) on which the first translation label is nonzero. Such a loop comes from lifting a product of a red and a blue meridian in the six-point pencil: its parity is even and its dihedral label is \(rb\), a nontrivial translation. The second label is zero on this factor. The loop can be represented in the good locus used in Lemma 19, since deleting its finite exceptional set is surjective on \(\pi_1(C)\) and the translation label extends over the filled curve.

Push \(\alpha\) to a loop \(e\) in \(O\) as in that lemma, so close to the distinguished fiber that its ball projection has diameter less than \(u\). Its basepoint has a central chart. The sum label on the pushed loop is the same nonzero integer as on \(\alpha\), by extension of the sum covering over the filled model. For every nonzero integer \(n\), the parametrized loop \(e^n\) has the same projected image as \(e\), and hence is itself one edge of \(G\).

The read endpoints of \(e^n\) are distinct in the central quotient graph. Before taking the symmetry quotient they differ by the nonzero translation \(n\,\ell(\alpha)\). The lift of \(e\) is closed in the actual ball and sign data, because the push is closed upstairs in \(V\) and its projection lies inside the prescribed small simply connected ball about \(o\). If the two translated endpoints became equal after quotienting by a lifted symmetry power, that power would fix their common ball point. This point lies in the ordinary open set where the symmetry acts freely, so the power would have to be the identity. A nonzero deck translation cannot fix a point of a covering. Thus the endpoints remain distinct.

By (36), every nonzero power of the class of \(e\) is nontrivial in \(\pi_1(\mathcal S)\). Lemma 19 puts this infinite-order element in the image of \(\pi_1(Y)\). That image is cyclic by Lemma 20; it is therefore infinite cyclic. This proves Theorem 18.

Replacing the orbifold ambient space

The projective-bundle construction below is a relative version of the free-locus method in the Godeaux–Serre construction; compare (Serre 1958, sec. 20) and (Totaro 1999, Remark 1.4 and §5). We include the containment and fundamental-group arguments, since the specified surface must survive the replacement.

Applied to \(Y\to\mathcal S\), this gives a smooth connected projective ambient variety \(S\) with an embedding of \(Y\) and \[ \mathop{\mathrm{im}}\bigl(\pi_1(Y)\longrightarrow\pi_1(S)\bigr)\simeq\mathbb Z. \tag{39}\] The remaining issue is that Proposition 2 requires \(A\) itself. We address one point blowup at a time.

Realizing a surface blowdown in a new ambient variety

The proof first places \(Y_1\) in a projective threefold and adjusts the normal bundle of its exceptional curve to \(\mathcal O(-1)\oplus\mathcal O(-1)\). Blowing up that curve gives an exceptional \(\mathbb P^1\times\mathbb P^1\). Contracting the other ruling then performs \(b\) on the surface. This is the classical local two-ruling construction; compare (Atiyah 1958). The essential global point here is to construct a polarization that makes this second contraction projective.

Completion of the construction