Surface groups are flexibly stable

We show that surface groups are flexibly stable in permutations. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.


Introduction
Let G = Σ|R be a finitely presented group with generators Σ and relations R. Let F Σ be a free group with basis Σ and π Σ : F Σ → G be the natural quotient map.
Consider the space Hom(F Σ , S N ) of homomorphisms from F Σ into the finite symmetric group S N for N ∈ N. We equip Hom(F Σ , S N ) with the normalized Hamming metric d N given by Definition. The group G is flexibly stable in permutations if for any ε > 0 there is δ = δ(ε) > 0 such that for any N ∈ N and ρ ∈ Hom(F Σ , S N ) satisfying |Fix(ρ(r))| = |{i ∈ {1, . . . , N } : ρ(r)(i) = i}| > (1 − δ)N ∀r ∈ R there exists M ∈ N with N ≤ M ≤ (1 + ε)N and ρ ∈ Hom(F Σ , S M ) that factors through π Σ and satisfies d M E M N (ρ), ρ < ε. Roughly speaking, flexible stability means that any map from G to a finite symmetric group sufficiently close to being a homomorphism is close to an actual homomorphism into a symmetric group of a possibly somewhat larger degree.
Flexible stability is independent of the particular finite presentation and as such is a property of the group G, see e.g. [BL18,§2] or [BLT18,3.13].
Theorem 1.1. Let S be a closed surface of genus g ≥ 2. The fundamental group is flexibly stable in permutations.
The third named author was partially supported by NSF grant DMS-1610827.
In fact our proof of Theorem 1.1 gives an explicit relation between ε and δ. Up to constants ε = δ ln(1/δ). This means that δ = o(ε) and ε 2 = o(δ) in the limit as both ε and δ go to 0.
On our method. Our approach to proving flexible stability is purely geometric. We use covering space theory to reformulate the problem in terms of certain branched covers of S we term * -covers. The goal then becomes making a given * -cover unramified by performing an amount of changes controlled by the total branching degree. First we cut the given * -cover along a carefully constructed embedded graph with geodesic edges. This results in surfaces that are embedded in covers of S and with combined boundary length controlled by the total branching degree.
We make sure the surfaces obtained as above have a locally convex boundary. Therefore they can be embedded in some cover of S with controlled area. This relies on the following result, which seems to be of independent interest. Theorem 1.2. Let S be a closed hyperbolic surface. Let R be a surface with boundary which is isometrically embedded in some cover of S. If the boundary ∂R is locally convex then R can be isometrically embedded in a cover Q of S such that where b S > 0 is a constant depending only on S.
The above can be regarded as a certain quantitative version of the LERF property for surface groups established by Scott [Sco78,Sco85]. As such it is closely related to and inspired by Patel's quantitative variant [Pat14] of Scott's theorem.
Unfortunately the methods used in the proof of Theorem 1.2 will in general increase the total area of X and for this reason we are only able to establish flexible rather than strict stability (a finitely presented group is said to be strictly stable in permutations if it satisfies the above definition of flexible stability with M being exactly equal to N ).
Stability and related works. There has been a significant recent interest in sofic groups [Wei00,ES06] as well as various notions of stability. A part of the interest in stability stems from the observation made in [GR09] that a non residually finite group which is stable in permutations cannot be sofic.
Groups known to be strictly stable in permutations include finite groups by Glebsky and Rivera [GR09], finitely generated abelian groups by Arzhantseva and Paunesco [AP15] and polycyclic and Baumslag-Solitar BS(1, n) groups by Becker, Lubotzky and Thom [BLT18].
Becker and Lubotzky [BL18] proved that a group G with Kazhdan's property (T) is not strictly stable by removing a single point from an action of G on a finite set. This strategy has led Becker and Lubotzky to introduce the flexible notion of stability in permutations and ask whether some Kazhdan groups might still be stable in the flexible sense. The question of strict and flexible stability for surface groups is discussed in [BL18] as well.
The fundamental group of a closed surface with constant non-negative curvature is either finite or abelian and as such stable by [GR09,AP15]. Therefore our Theorem 1.1 completes the picture for all closed surfaces.
Free non-abelian groups are clearly stable in a void sense. On the other hand there exist finitely presented hyperbolic groups that are not even flexibly stable [BM18]. Interestingly, while hyperbolic surface groups are possibly the simplest example of non-free hyperbolic groups, the question of stability for these groups is not trivial. To the best of the authors' knowledge it is not known whether surface groups are stable in permutations in the strict sense. The problem of adapting our present approach to deal with this question seems challenging.
Finally we point out that it was recently shown by Becker and Mosheiff [BM18] that for the free abelian group Z d the parameter δ = δ(ε) goes to zero at least as fast as a polynomial of degree d in ε. So in some quantitative sense hyperbolic surface groups are "more stable" than free abelian ones. * -covers of surfaces. Let S be a closed surface. Endow S with a metric of constant sectional curvature. Recall that a finite branched cover of S is a continuous surjection p : X → S such that every point b ∈ S admits a neighborhood b ∈ V b ⊂ S with p −1 (V b ) = U 1 · · · U n b and such that f |Ui is topologically conjugate to the complex map z → z d of some degree d ∈ N depending on i. The degree is equal to one unless b belongs to a finite subset of S called the branch set.
Definition. A * -cover of S is a compact surface X admitting a branched cover p : X → S with branch set consisting of a single branch point * ∈ S.
Let p : X → S be a * -cover. In general X is not required to be connected. We pull back the metric from S to every connected component of X so that p is a local isometry away from p −1 ( * ).
The total angle α x locally at any point x ∈ X is an integer multiple of 2π. In The singular set of X is s(X) = {x ∈ X : d x > 1}. Clearly s(X) ⊂ p −1 ( * ) and in particular s(X) is discrete. The branching degree of X is β(X) = x∈s(X) (d x − 1). We say that the * -cover X is unramified if s(X) = ∅, or equivalently if β(X) = 0. This happens if and only if p is a covering map. We emphasize that covers are not required to be connected.
The degree |X| of the * -cover p : X → S is equal to |p −1 (x)| for any x ∈ S \ { * }.
Geometric stability. The geometric notion of stability is defined in terms of certain graphs embedded into * -covers. Recall that a graph is a simplicial 1-complex. We let Γ (0) denote the vertex set of the graph Γ.
Definition. A * -graph on X is an embedded graph Γ such that Γ (0) ⊂ p −1 ( * ) and the edges of Γ are geodesic arcs.
The fact that Γ is embedded means that two edges of Γ may intersect only at the vertex set Γ (0) . If Γ is a * -graph on X then any closed curve γ contained in Γ is a piece-wise geodesic closed curve in X. Given a * -graph Γ let l(Γ) denote the total length of all of its edges.
Definition. The * -cover p : X → S is ε-reparable if X admits a * -graph Γ with l(Γ) ≤ εArea(X) such that the complement X \ Γ isometrically embeds into a cover C of S satisfying We emphasize that the complement X \ Γ as well as the cover C are in general allowed to be disconnected.
Our main result can be reformulated in the language of * -covers.
Theorem 2.1. A closed hyperbolic surface is flexibly geometrically stable.
In the remaining part of this section we show that geometric stability implies algebraic stability in permutations.
Permutation representations and covering theory. Consider the punctured Covering space theory [Hat02,p. 68] shows how to associate a natural permutation representation ρ X : F → S |X| to the covering X \ p −1 ( * ) → S \ { * }. Note that ρ X is well-defined up to an inner automorphism of S |X| . In fact the permutation representation ρ X is naturally acting on the set In what follows it is convenient to fix a specific presentation for the fundamental group of S. Since the notion of flexible stability is known to be independent of the chosen presentation we may do so without any loss of generality.
Let P be a compact fundamental domain for the action of the fundamental group π 1 (S, x 0 ) on the universal cover S. Assume that P is a polygon with finitely many geodesic sides and that the vertices of P are all lifts of the point * ∈ S. Moreover assume that the lift of the point x 0 lies in the interior of P.
Let Σ be the finite generating set of π 1 (S, x 0 ) consisting of a single element σ from any pair {σ, σ −1 } ⊂ π 1 (S, x 0 ) such that σP ∩ P is a geodesic side of P. We may identify F with the free group F Σ on the generators Σ. There is a natural quotient homomorphism π Σ : F Σ → π 1 (S, x 0 ). The surface group π 1 (S, x 0 ) admits a presentation with generating set Σ and a single relation r ∈ F Σ .
In particular we may let the fundamental domain P be a 4g-sided polygon so Proposition 2.2. Given any ρ ∈ Hom(F Σ , S N ) with N ∈ N there exists a * -cover X ρ → S with |X ρ | = N and ρ Xρ = ρ. Moreover β(X ρ ) ≤ N − |Fix(ρ(r))| where r is the defining relation of π 1 (S, x 0 ) as above.
Proof. Let X ρ be the punctured surface covering S \ { * } that corresponds to the permutation representation ρ : F Σ → S N . Let p : X ρ → S be the * -cover obtained by completing X ρ at the punctures. It is clear that |X ρ | = N and ρ Xρ = ρ.
Consider the cycle decomposition o 1 · · · o m of the permutation ρ(r) in its action on the set p −1 (x 0 ) ∼ = {1, . . . , N }. Let l i denote the length of the cycle o i . We claim that without loss of generality p −1 ( * ) = {y 1 , . . . , y m } and d yi = l i . This would imply the required upper bound on β(X ρ ) as Let γ be a simple closed curve in S \{ * } based at x 0 and representing the element r of F Σ . The preimage of the curve γ in X ρ is a disjoint union of simple closed curves. Every such curve corresponds to an orbit o i of ρ(r) in its action on p −1 (x 0 ) and bounds a disc in X ρ that contains a single point y i from p −1 ( * ) in its interior. Moreover the degree d yi is equal to the size l i of the orbit o i . The claim follows.
Geometric stability implies algebraic. We show that geometric stability in the sense of Theorem 2.1 implies our main result Theorem 1.1. We continue using the presentation π 1 (S, x 0 ) ∼ = Σ|r constructed above and in particular the polygon P.
Proposition 2.3. Let Γ be a * -graph on X such that the complement X \ Γ isometrically embeds into some cover C of S of degree M = |C|. Then where N = |X| and a S > 0 is a constant depending only on the surface S. Proof of Proposition 2.3. The map p : X → S induces a tessellation T = T (X, P) of the * -cover X by |X|-many isometric copies of the polygon P. We claim that where a S > 0 is a constant depending only on the surface S, the choice of the fundamental domain P and the particular generating set Σ.
To establish the claim consider a polygon D belonging to the tessellation T . Recall that p −1 ( * ) ∩ D is the set of vertices of D and consider any connected component α of (D ∩ Γ) \ p −1 ( * ). In particular α is a geodesic arc. If α is contained in a geodesic side of D then α must be equal to that side.
Otherwise the arc α has a non-trivial intersection with the interior of D. There is a definite lower bound on the length of any geodesic arc contained in the interior of P unless it comes within some definite small distance from the vertex set of P. On the other hand, there is an upper bound on the number of polygons from the tessellation T admitting a non-trivial intersection with a geodesic arc contained in a small neighborhood of some vertex of the tessellation T . This bound depends on P but is independent of the degree of the vertex in question. The claim follows from the above considerations.
Let p denote the covering map from C to S. We may identify p −1 (x 0 ) with a subset of p −1 (x 0 ) and regard the two permutation representations ρ X and ρ C respectively as acting on these two sets. Consider a point x ∈ p −1 (x 0 ) and let D ∈ T be the polygon containing x in its interior. Given a particular generator σ ∈ Σ let D σ ∈ T be the polygon sharing with D the geodesic side corresponding to σ. Then E M N (ρ X )(σ)(x) = ρ C (σ)(x) provided that the * -graph Γ does not intersect (D ∪ D σ ) \ p −1 ( * ). The Hamming metric between E M N (ρ X ) and ρ C is therefore bounded above by and the proof follows for an appropriate choice of the constant a S > 0.
Proposition 2.4. Let S be a closed hyperbolic surface. If S is flexibly geometrically stable then its fundamental group π 1 (S, x 0 ) is flexibly stable in permutations.
Proof. The algebraic property of flexible stability in permutations does not depend on the choice of a particular presentation. This is proved in [BL18,§2]  Exactly the same argument goes through in the flexible case as well. It will be convenient to verify the definition with respect to the presentation π 1 (S, x 0 ) ∼ = Σ|r . Let ε > 0 be given. Denote ε = min{ε, ε 2a S }. According to our assumption there is some δ = δ(ε ) such that any * -cover X with β(X) < δArea(X) is εreparable. We claim that the definition of flexible stability in permutations for the group π 1 (S, x 0 ) is satisfied with respect to the given ε > 0 and this particular δ.
Consider the permutation representation ρ = ρ C so that ρ ∈ Hom(F Σ , S M ) where M = |C|. Since C is unramified the homomorphism ρ factors through π Σ : F Σ → π 1 (S, x 0 ). Roughly speaking, the two permutations representations ρ and ρ agree on points lying on the subsurface X \ Γ of C away from the * -graph Γ. To be precise it follows from Proposition 2.3 that The remainder of this work is dedicated to establishing Theorem 2.1.

Hyperbolic planes with singularities
Let S be a compact hyperbolic surface and consider a fixed * -cover p : X → S. We discuss the geometry of the universal cover of X. We then discuss the Voronoi tessellation and its dual the Delaunay graph. This graph will be used in Section 4 to construct the * -graph as required in the definition of flexible geometric stability.
The geometry of the universal cover of X. Let q : X → X denote the universal cover of X equipped with the pullback length metric d X . Topologically speaking X is homeomorphic to the plane.
The singular set of X is s( X) = q −1 (s(X)). The space X is locally isometric to the hyperbolic plane H away from its singular set. The group π 1 (X) acts freely on X admitting the surface X as a quotient. The singular set s( X) is discrete in X. In fact s( X) is co-bounded provided s(X) is not empty.
Recall that the Cartan-Hadamard theorem admits a generalization due to Gromov to complete geodesic metric spaces. See [BH13, Theorem II.4.1] for reference. This result implies that X is a CAT(−1)-space. In particular X is uniquely geodesic and every local geodesic in X is a geodesic.
Let γ be a continuous path in X. The path γ is a local geodesic provided it is a local geodesic in the sense of hyperbolic geometry away from the singular set s( X) and its angle θ x at every singular point x ∈ s( X) along γ satisfies π ≤ θ x ≤ α x − π where α x is the local angle at x. This local characterization implies that X is geodesically complete in the sense that any geodesic segment can be extended to a bi-infinite geodesic line. The geodesic extension need not be unique since a geodesic segment terminating at a singular point admits many extensions.
Convex subsets of the universal cover X. Recall the following useful consequence of the Cartan-Hadamard theorem [BH13, II.4.13 and II.4.14].
Lemma 3.1. Let N 1 and N 2 be connected complete non-positively curved metric spaces and f : N 1 → N 2 be a local isometry. Then f * : π 1 (N 1 ) → π 1 (N 2 ) is injective and any lift F : N 1 → N 2 of the map f is an isometric embedding.
Here is an example of a straightforward application of Lemma 3.1.
Corollary 3.2. Let C ⊂ X be a convex subset such thatC ∩ s( X) = ∅. Then C is isometric to a contractible subset of the hyperbolic plane H.
Proof. Since C is a convex subset of X it is a CAT(−1)-space in its own right. The corollary follows by applying Lemma 3.1 with N 1 = C, N 2 = S and f = p • q. The lift F gives the required isometric embedding into H.
Let γ be a bi-infinite geodesic path in X. A half-space in X is the closure of a connected component of X \ γ. Note that a half-space is convex.
Voronoi cells and the Delaunay graph. Assume that the singular set s(X) is non-empty. We define the Voronoi cells and the Delaunay graph with respect to the set of points s( X). These are natural generalizations of the parallel notions in the classical Euclidean and hyperbolic cases.
The family of Voronoi cells is equivariant in the sense that gA v = A gv for every g ∈ π 1 (X) and v ∈ s( X).
We remark that it is not a priori clear that the Voronoi cells form a tessellation. This turns out to be true and will be established as a consequence of Proposition 3.4 below. The difficulty has to do with the fact that for two singular points v, u ∈ s( X) the intersection of the two sets {x ∈ X : Definition. The vertex set of the Delaunay graph D is the singular set s( X). Two vertices v 1 and v 2 of the Delaunay graph D span an edge whenever there is a closed We summarize a few basic properties of Voronoi cells and their relationship with the Delaunay graph. These are quite elementary in the classical Euclidean or hyperbolic cases. Extra caution is required in the presence of singular pointssome of the following statements are false unless the Voronoi cells are taken with respect to a set of points containing all singularities, which is always the case here.
The proofs of Propositions 3.3, 3.4, 3.6 and 3.7 are postponed to Appendix A dedicated to this topic. The proofs are similar to the classical case by repeatedly relying on Corollary 3.2 to embed the relevant local picture into the hyperbolic plane.
where u ∈ s( X) ranges over the vertices adjacent to v in the graph D.
The following proposition describes the intersection of two Voronoi cells.
Proposition 3.4. Let v, u ∈ s( X) be a pair of distinct vertices of the Delaunay graph. Consider the two Voronoi cells A v and A u . As a consequence of Proposition 3.4 we deduce that the family of Voronoi cells A v for v ∈ s( X) forms a tessellation of X called the Voronoi tessellation. Moreover the Delaunay graph D is dual to this tessellation. These facts are well-known in the classical Euclidean and hyperbolic situations.
Corollary 3.5. Let v ∈ s( X) be a vertex of the Delaunay graph. Then the interior of the Voronoi cell A v embeds into X via the restriction of the covering map q : The geometric realization of the Delaunay graph. From now on we regard the Delaunay graph D as being geometrically realized in X. More precisely, we identify the vertices of D with the singular set s( X) and realize every edge of D by the corresponding geodesic arc in X.
Proposition 3.6. The Delaunay graph D is embedded in X. The projection q(D) is embedded in the surface X.
The above statement means that geodesic arcs realizing two distinct edges of D may intersect only at a vertex incident to both. In particular D is planar. Likewise two distinct edges of q(D) may intersect only at a common singular point of s(X).
Proposition 3.7. Any connected component of the complement of the Delaunay graph in X is locally convex.
An estimate for the area of Voronoi cells. Recall that a subset L of a metric space M is r-separated if d M (x 1 , x 2 ) ≥ r for any two distinct points x 1 , x 2 ∈ L. Moreover recall that we defined a half-space in X to be the closure of a connected component of the complement of some bi-infinite geodesic line in X.
where ϕ > 0 is a constant depending only on the distance r.
The expression appearing on the right-hand side of the above estimate is the area of the hyperbolic sector with central angle ϕ and of radius R/2. Throughout the following proof and given a point p ∈ X it is convenient to introduce the notation Proof of Lemma 3.8. Let us first determine the angle ϕ > 0 as follows. Let l 1 and l 2 be a pair of geodesic lines in the hyperbolic plane H such that d H (l 1 , l 2 ) = r. Let m be the mid-point of the geodesic arc perpendicular to both l 1 and l 2 . Then ϕ is the angle between the two geodesic rays emanating from the point m towards the ideal points l 1 (∞) and l 2 (∞).
Let W R r be the subset of X given by . As a consequence of Corollary 3.2 the two convex sets B X (v, r) and H ∩ B X (v, R) are isometric to a hyperbolic ball of radius r and a hyperbolic sector of radius R and angle π, respectively. Let Q ϕ,R/2 be the hyperbolic sector of angle ϕ and radius R/2 which is based at the vertex v and contained in W R r . The lemma will be established by showing that W R r and therefore Q ϕ,R/2 is contained in H ∩ A v . Let u 1 , . . . , u n with n ∈ N be the vertices of the Delaunay graph D adjacent to the vertex v. For every i ∈ {1, . . . , n} let w i be the point along the geodesic arc Claim. Let u i be one of the Delaunay vertices adjacent to v.
Proof of Claim. Consider a point x ∈ W R r . We need to show that x ∈ F(w i ). Note that w i and x lie on opposite sides of the bi-infinite geodesic ρ. Both w i and x belong to the following locally convex and contractible and hence convex set Therefore the intersection point z of ρ with the geodesic arc from w i to x lies along the geodesic arc from v −r to v r . This relies on the fact that r ≤ R as follows from the assumptions and on the observation that Assume without loss of generality that in fact the intersection point z lies along the geodesic arc from v to v r . Consider the two geodesic triangles The triangles T 1 and T 2 have no singular points in their interior. Therefore T 1 and T 2 are isometric to their respective comparison hyperbolic triangles according to Observe that the respective angles of T 1 and T 2 at the vertices w i and v r satisfy On the other hand the assumptions of Lemma 3.8

Cut graphs on branched covers
Let S be a compact hyperbolic surface. Consider an arbitrary * -cover X of S. To prove that S is flexibly geometrically stable we need to construct a * -graph Γ on X so that the complement X \ Γ embeds into some cover of S. In addition the total length of Γ should be controlled by the branching degree of X when both quantities are normalized relative to the size of X.
Definition 4.1. Let X be a * -cover of S. A c-cut graph for some c > 0 is a * -graph Γ on X satisfying (1) the singular set s(X) is contained in the vertex set Γ (0) , (2) if e 1 and e 2 are two edges incident at the vertex x ∈ Γ (0) and consecutive in the cyclic order on the link of graph Γ at the vertex x induced by its embedding in X then the angle between e 1 and e 2 at x is at most π, and (3) the total edge length l(Γ) is bounded above by cArea(X). A cut graph on X is a c-cut graph for some c > 0.
The above Condition (2) is equivalent to saying that every connected component of X \ Γ has locally convex boundary. The main goal of the current section is the following.
Roughly speaking, the graph Γ will be constructed by considering the Delaunay graph and then making it more sparse by carefully removing some of its edges. The function c will depend on the topology and the metric of the surface S.
Proof of Theorem 4.2. Let p : X → S be a * -cover whose branching degree satisfies β(X) < δ|X| for some sufficiently small δ > 0 to be determined below. Recall that the branching degree β(X) is defined to be v∈s( We may assume that the singular set s(X) is non-empty for otherwise the empty graph is a c-cut graph for any c > 0. Consider the singular set s( X) = q −1 (s(X)). Since s( X) ⊂ (p • q) −1 ( * ) this set is r-separated for some constant r > 0 depending only on the injectivity radius of the compact surface S.
Consider the family of Voronoi cells in X with respect to the vertex set s( X). The interior of the every cell A v with v ∈ s( X) embeds into X according to Corollary 3.5. Denote A v = q(A v ) where v ∈ s(X) and v is any vertex in q −1 (v). The family of the A v 's with v ∈ s(X) forms a tessellation of X and satisfies Recall that a half-space in X is the closure of a connected component of the complement of some bi-infinite geodesic line in X.
Let D be the geometric realization of the Delaunay graph in X with respect to the vertex set s( X). This notion is discussed in Section 3 above. It follows from Proposition 3.6 that q(D) is a * -graph in X. This * -graph satisfies Condition (1) by its construction and Condition (2) according to Proposition 3.7. The required cut graph will be obtained by discarding some of the edges of q(D) while making sure Condition (2) continues to hold.
Let L v denote the set of edges of the graph q(D) incident at the vertex v ∈ s(X). Our next step is to find a subset M v of L v of size |M v | ≤ 4d v that continues to satisfy Condition (2) of cut graphs and such that where ϕ > 0 is the constant given in Lemma 3.8 with respect to the parameter r.
Consider the subset K v of L v consisting of all edges e such that 2ϕ sinh 2 (l(e)/4) ≤ Area(C e ) holds for some half-cell C e at v containing the edge e in its interior. We claim that K v satisfies Condition (2) of cut graphs. If this was not the case then there would exist a half-cell C at v which does not contain any edge of K v in its interior. However at least one edge e ∈ L v is contained in C and every such edge is therefore not in K v . Therefore 2ϕ sinh 2 (l(e)/4) > Area(C) for every edge e ∈ L v contained in the half-cell C. This stands in contradiction to Lemma 3.8. Take M v to be a subset of K v minimal with respect to containment that still satisfies Condition (2) of cut graphs. The minimality of M v implies that any halfcell C at v contains at most two edges from M v . Since A v is the union of 2d v half-cells at v it follows that |M v | ≤ 4d v . Moreover without loss generality any point x of the Voronoi cell A v can be regarded as being on the boundary of two of these 2d v half-cells at v covering A v , and therefore x belongs to at most four half-cells from the family {C e } e∈Mv . Equation (II) follows.
Consider the sub- * -graph Γ embedded in X with vertex set s(X) and edge set consisting of v∈s(X) M v . Observe that Conditions (1) and (2) of cut graphs hold true. It remains to bound the total length l(Γ) from above as in Condition (3) of cut graphs and to determine the precise value of δ.
Let E denote the set of edges of Γ. To conclude the proof define the function c : δ → c δ by Note that c is monotone and that lim δ→0 c δ = 0. Finally observe that
Therefore Γ is a c-cut graph provided δ is sufficiently small so that c δ < c.

Boundaries and non-separating closed curves
Let S be a compact hyperbolic surface. We discuss the existence of certain covers of surfaces with boundary such that the boundary curve maps to a non-separating closed curve on S.
The following two Lemmas 5.1 and 5.2 are both special cases of the more general [Neu01, Lemma 3.2] due to Walter D. Neumann. We have chosen to include the proofs for the sake of the completeness. The proof found in [Neu01] is considerably more condensed.
Lemma 5.1. Let α be a non-separating oriented simple closed curve on S. There exists a compact subsurface R embedded in a cover of S whose boundary has p + q components mapping to α r1 , . . . , α rp and α −l1 , . . . , α −lq with respect to the induced orientation for some numbers r 1 , . . . , r p ∈ N and l 1 , . . . , l q ∈ N if and only if there is some N ∈ N such that p i=1 r i = q j=1 l j = N and p + q is even.
Moreover R can be taken to be embedded in a cover of degree 2N .
The cover of S in the above statement can possibly be disconnected (i.e. an unramified * -cover in our terminology). Note that assuming the first condition, the second condition is equivalent to Proof of Lemma 5.1. The non-separating simple closed curves on S are all topologically indistinguishable and so the fundamental group of S admits a particular presentation where a 1 ∈ π 1 (S) is the element representing the curve α. Pick a basepoint x 0 ∈ S lying on the curve α and a system of simple closed curves α 1 = α, β 1 , . . . , α g , β g based at x 0 and representing the generators such that S \ (α 1 ∪ β 1 ∪ · · · ∪ α g ∪ β g ) is a disc. Let F be a lift of that disc which is a fundamental domain for the action of π 1 (S) on its universal cover. The boundary of F reads α 1 β 1 α −1 1 β −1 1 · · · α g β g α −1 g β −1 g . Allowing for a slight abuse of notation, we regard x 0 as being a particular point on the boundary of the disc F by identifying it with the point incident to the edges α 1 and β −1 g . Covers of S and permutation representations of π 1 (S). As in [Hat02, p. 68] covers of S correspond to actions of the group π 1 (S) on some set Ω by permutations. The degree of the cover is equal to |Ω|. The connected components of the cover correspond to the orbits of π 1 (S) in Ω.
Consider a given cover Q of S. The system of curves α 1 , β 1 , . . . , α g , β g lifts to an embedded graph 1 Γ in Q whose vertex set p −1 (x 0 ) can be identified with the set Ω. The faces of the graph Γ on the surface Q correspond to polygons of the tessellation T (Q, F). The boundary of each face reads the defining relation in the above presentation.
Edges of Γ labeled α 1 appear once along the boundary of every face of T (Q, F) with opposite orientations. Any two such edges are lifts of α to Q based at points of Ω differing by an action of the generator b 1 .
We construct a surface R embedded in a cover of degree 2N given a set of numbers as in the statement. Consider an action of π 1 (S) on the disjoint union Ω of two sets Ω + and Ω − of size N each, as follows. Let a 1 act as a permutation of Ω consisting of p cycles of sizes r 1 , . . . , r p on the set Ω + and q cycles of sizes l 1 , . . . , l q on the set Ω − . Let b 1 act by any involution with b 1 (Ω + ) = Ω − . The sets Ω + and Ω − are a 1invariant as well as [a 1 , b 1 ]-invariant. The assumption implies that the signatures of the restrictions a 1|Ω+ and a 1|Ω− are the same. Hence the signature of [a 1 , b 1 ] |Ω+ as well as that of [a 1 , b 1 ] |Ω− is +1.
By [Ore51, Theorem 1] every even permutation can be written as a commutator. Therefore there exists a permutation action of a 2 and b 2 which leaves the sets Ω + and Ω − invariant and satisfies [a 1 , b 1 ][a 2 , b 2 ] = id Ω . If g > 2 let the generators a i and b i act as identity on Ω for all i ∈ {3, . . . , g}.
The above permutation representation of π 1 (S) determines a cover p : Q → S of degree 2N . Consider the subsurface R of Q given by the union of all lifts to Q of the polygon F with base point x 0 lifted to these points of p −1 (x 0 ) corresponding to the set Ω + . It remains to verify R has the desired properties.
Any lift of the polygon F contained in R has all of its vertices belonging to Ω + , except for two vertices that belong to Ω − and are the terminal points of lifts of β 1 into the boundary of F. The same statement applies to any lift of F not contained in R reversing the roles of Ω + and Ω − . Therefore the intersection of a small neighborhood of any vertex of the graph Γ with the subsurface R is completely determined by whether that vertex belongs to Ω + or to Ω − . It follows that the boundary components of R exactly correspond to the powers of α as stated.
Conversely, suppose that R is a compact subsurface embedded in a cover p : Q → S whose boundary has p + q components mapping to α r1 , . . . , α rp , α −l1 , . . . , α −lq . We show that these numbers satisfy the conditions of the statement.
The boundary of R is contained in the union of all edges of Γ so that R is a union of polygons from T (Q, P). Let Ω denote the vertex set of Γ. Let Ω + be the subset of Ω consisting of all points x ∈ p −1 (x 0 ) such that the lift of the polygon F to Q with basepoint x 0 being lifted to x is contained in R.
The group π 1 (S) is acting on the set Ω by permutations. Observe that Ω + is invariant under both a 1 as well as b 1 a −1 1 b −1 1 . In particular Ω + is [a 1 , b 1 ]-invariant. In addition Ω + is [a i , b i ]-invariant for all i ∈ {2, . . . , g}. The defining relation of π 1 (S) implies that the signature of [a 1 , b 1 ] |Ω+ is +1, or equivalently that the signatures of the two permutations a 1|Ω+ and (b 1 a −1 1 b −1 1 ) |Ω+ coincide. Consider the cycle decompositions of the two permutations a 1 and b 1 a −1 1 b −1 1 restricted to Ω + . These decompositions are identical except for the cycles lying on the boundary of R. More precisely, boundary components mapping to α with positive orientation give rise to orbits of a 1|Ω+ but not of (b 1 a −1 1 b −1 1 ) |Ω+ , while boundary components mapping to α with negative orientation correspond to orbits of (b 1 a −1 1 b −1 1 ) |Ω+ but not of a 1|Ω+ . The statement involving the product of these signatures follows.
The fact that the sum of the r i is equal to the sum of the l j follows from a simple homological computation, see e.g. Proposition 6.4 below.
We will require another closely related lemma dealing with boundaries mapping to two disjoint non-separating curves.
Proof. We proceed in a similar manner to the proof of the forward direction of Lemma 5.1. All pairs of disjoint non-separating simple closed curves on S are topologically indistinguishable and so the fundamental group of S admits a presentation where a 1 and b 2 are the elements of π 1 (S) representing the curves α and β respectively. Let γ be a simple arc on S connecting α and β. Let x 0 be the intersection point of the arc γ with the curve α. Pick a system of closed curves α 1 = α, β 1 , α 2 , β 2 = β, . . . , α g , β g based at x 0 and representing the generators such that S \ (α 1 ∪ β 1 ∪ · · · ∪ α g ∪ β g ∪ γ) is a disc. Let F be a fundamental domain for the action of π 1 (S) on its universal cover which is a lift of that disc.
By a slight abuse of notation, we regard x 0 as being a particular point on the boundary of the disc F by identifying it with the point incident to both α 1 and γ.
We rely on the correspondence between covers of S and permutation representations of π 1 (S) as in Lemma 5.1. Consider an action of π 1 (S) on the disjoint union Ω of two sets Ω + and Ω − of size N each, as follows. Let a 1 and b 2 act by the same permutation of Ω that consists of p cycles of sizes r 1 , . . . , r p on the set Ω + and q cycles of sizes l 1 , . . . , l q on the set Ω − . Let b 1 and a 2 act by the same involution with b 1 (Ω + ) = a 2 (Ω + ) = Ω − . The two sets Ω + and Ω − are a 1 and b 2 -invariant as well as [a 1 , b 1 ] and [a 2 , b 2 ]-invariant. Observe that [a 1 , b 1 ] = [b 2 , a 2 ] = [a 2 , b 2 ] −1 and so [a 1 , b 1 ][a 2 , b 2 ] = id Ω . If g > 2 then let a i and b i be the identity on Ω for all i ∈ {3, . . . , g}.
The above permutation representation of π 1 (S) determines a cover p : Q → S of degree 2N . Consider the subsurface R of Q given by the union of all lifts to Q of the polygon F with base point x 0 being lifted to these points of p −1 (x 0 ) corresponding to the subset Ω + . Arguments analogous to Lemma 5.1 now show that R has the desired properties.

Quantitatively capping off surfaces with boundary
Let S be a closed hyperbolic surface of genus g so that g ≥ 2. The goal of the current section is to prove Theorem 1.2, restated below for the reader's convience.
Theorem. Let R be a surface with boundary which is isometrically embedded in some cover of S. If the boundary ∂R is locally convex then R can be isometrically embedded in a cover Q of S such that where b S > 0 is a constant depending only on S.
We will deal with the problem of capping-off a boundary component of R in a controlled way by working in a certain combinatorial framework. Namely we will consider surfaces tessellated by isometric copies of a particularly nice fundamental domain. The generic situation as in the above theorem can be easily reduced to this combinatorial framework.
Surfaces with locally convex boundary. We point out the following consequence of Lemma 3.1 and of the Cartan-Hadamard theorem.
Lemma 6.1. Let R be a surface with boundary. Then the boundary ∂R is locally convex and R is isometrically embedded in some cover of S if and only if R admits a local isometry to S.
Proof. If R is isometrically embedded in some cover p : C → S and its boundary ∂R is locally convex then the restriction of p to R is a local isometry.
Conversely assume that R admits a local isometry f into S. This implies that the boundary ∂R is locally convex. The map f * : π 1 (R) → π 1 (C) is injective and f lifts to an isometric embedding F : R → S ∼ = H of the universal covers according to Lemma 3.1. The map F descends to an isometric embedding of R into the cover of S that corresponds to the subgroup f * π 1 (R) ≤ π 1 (S).
Let P denote the completion of the complement S \ (λ 0 ∪ · · · ∪ λ 2g−1 ). So P is compact convex hyperbolic polygon. Moreover P is isometric to a fundamental domain for the action of the fundamental group π 1 (S) on the hyperbolic plane. We introduce the following notations for the edges of P; see Figure 2.
• e 0 andē 0 are the two geodesic sides of P that correspond to the curve λ 0 , • e 2g−1 andē 2g−1 are the two geodesic sides of P that correspond to the curve λ 2g−1 , and • e i , e i ,ē i andē i are the four sides of P that correspond to the curve λ i for every other i ∈ {1, . . . , 2g − 2}. The surface S can be recovered by identifying every geodesic side e of the fundamental polygon P with the geodesic side of opposite orientation denotedē.
We will keep the above notations throughout the remainder of Section 6.
Tessellations and P-surfaces. Let T (C, P) denote the induced tessellation of a given cover C of the surface S by isometric copies of the fundamental polygon P.
The link of every vertex of the tessellation T (C, P) has size four. In other words, exactly four polygons of T (C, P) meet at every vertex. Another crucial property of the tessellation T (C, P) is that the concatenation of any pair of edges incident at a given vertex v and not consecutive in the cyclic ordering determined by the link of v is a local geodesic. Such a concatenation of two edges is, up to orientation, of the form e 0 e 0 , e 2g−1 e 2g−1 , e i e i or e i e i for some i ∈ {1, . . . , 2g − 2}.
Definition. A P-surface R is a subsurface of some cover C of the surface S tiled by polygons from the tessellation T (C, P). The induced tessellation of R will be denoted T (R, P).
We remark that a P-surface could in general be disconnected. Proposition 6.3. Let R be a P-surface. Then every geodesic arc e of ∂P appears along the boundary ∂R the same number of times with each orientation e and e.
Proof. Every geodesic arc e appears along the boundary of the polygon P once with each orientation. Moreover in the tessellation of R by isometric copies of P every interior edge is accounted for once with each orientation. The result follows.
Let R be a P-surface. It follows from the properties of the tessellation T (R, P) discussed above that ∂R is locally convex if and only the link of any vertex v of T (R, P) that lies on the boundary ∂R has size at most two. This observation motivates the following.
Proposition 6.4. Let R be a P-surface with locally convex boundary. Let α and β be two totally geodesic closed curves/maximal geodesic arcs along the boundary ∂R that map to a geodesic curve/arc on S with opposite orientations. Then the surface R obtained by identifying α and β is a P-surface with locally convex boundary.
Proof. The fact that R has a locally convex boundary is clear in the case that α and β are totally geodesic boundary curves.
Consider the case where α and β are maximal geodesic arcs. The vertices representing the end-points of the arcs α and β in the tessellation T (R, P) have links of size one in R. Therefore the boundary components of R that have been modified by the identification of α and β have links of size at most two in R and so remain locally convex as required.
The P-surface structure of R gives rise to a local isometry f : R → S. It is compatible with the identification of α and β and descends to a local isometry f : R → S. It follows from Lemma 6.1 that R isometrically embeds into some cover of S. It is clear that R is tiled by polygons from the tessellation T (C, P). Therefore R is a P-surface.
The notion of P-surfaces is useful for our purposes since a surface with boundary as in Theorem 1.2 can always be embedded inside a P-surface in an efficient way.
Proposition 6.5. Let R be a surface with boundary which is isometrically embedded in some cover of S. If the boundary ∂R is locally convex then R can be isometrically embedded in a P-surface Q with locally convex boundary such that where d = d S > 0 is a constant depending only on the surface S.
Moreover we may assume, if desired, that any geodesic subarc of ∂Q has at most two edges and that no boundary component of ∂Q is totally geodesic.
Proof. We assume that R has non-empty boundary for otherwise there is nothing to prove. Since R is embedded in some cover C of S we may identify π 1 (R) with a certain infinite index subgroup of π 1 (S). Let R be the cover of S corresponding to that subgroup. In particular R is an embedded subsurface of R and every connected component of R \ R retracts to a boundary component of ∂R. Let Q be the P-surface consisting of all polygons in T (R , P) that intersect R non-trivially.
The boundary of the P-surface Q need not yet be locally convex. Patel's argument in [Pat14, Theorem 3.1] shows that there is a P-surface Q with locally convex boundary and Q ⊂ Q ⊂ R that can be obtained by attaching extra polygons to Q . These new polygons are attached only along the boundary of Q .
We emphasize that in [Pat14, Theorem 3.1] the polygon P is a right-angled regular pentagon. However Patel's proof relies only on certain properties of links in the tessellation T (R , P ) that were discussed above and hold true in our case as well. Therefore the same proof goes through.
Finally, the boundary of the P-surface Q is locally convex but may contain geodesic arcs longer than two edges or totally geodesic boundary components. If desired, this can be overcome simply by attaching an additional layer of polygons to Q . Namely, let Q consist of all polygons in T (R , P) that intersect Q or its boundary non-trivially. The boundary of Q remains locally convex and satisfies the requirements as in the statement.
An area estimate analogous to [Pat14,Theorem 4.3] shows that Area(Q \ R) ≤ d S l(∂R) for some constant d S > 0. The total boundary length of Q is bounded above by the number of polygons meeting Q \ R times the perimeter of the polygon P. In particular l(∂Q ) ≤ d S l(∂R) for some other constant d S > 0. Repeating this argument twice for the pair of P-surfaces Q and Q gives the required linear upper bounds on Area(Q \ R) and l(∂Q) in terms of some positive constant d S > 0.
Capping off boundary components. We now have all the required machinery to complete the proof of Theorem 1.2 of the introduction.
Proof of Theorem 1.2. Let R be a surface with locally convex boundary which is embedded in some cover of S. Making use of Proposition 6.5 it is possible to find a P-surface Q 1 containing an embedded copy of R such that Area(Q 1 \ R) as well as l(∂Q 1 ) are bounded above by d S l(∂R) where d S > 0 is a positive constant. Moreover any geodesic subarc of ∂Q 1 has at most two edges and no component of ∂Q 1 is geodesic. Recall that P is the convex polygonal fundamental domain obtained as the complement of the system of curves λ 0 , . . . , λ 2g−1 on the surface S.
The boundary components of Q 1 are piecewise geodesic and map to the arcs e 0 , . . . , e 2g−1 . Each geodesic boundary arc of length one is labelled by either e i , e i or their inverses. Each arc of length two is labelled by either e 0 e 0 , e 2g−1 e 2g−1 , e i e i , e i e i or their inverses. We say that a geodesic boundary arc has odd label or even label if i is odd or even, respectively.
The proof proceeds in four steps.
(1) Identifying length two geodesic boundary arcs with even labels. Consider a geodesic boundary arc α of length two and even label. We will assume that with the induced orientation α reads e i e i with i even. The other cases are treated analogously. The midpoint of the length two arc α has an incident edge contained in the interior of Q 1 and labeled e i+1 . Trace a geodesic arc γ on the surface Q 1 starting at this edge with labels alternating between e i+1 and e i+1 . It runs transverse to geodesic arcs of the two forms e i e i and e i+2 e i+2 until it reaches the boundary of Q 1 again, necessarily at the midpoint of some length two geodesic arc.
There are two possibilities to consider. If γ reaches the boundary of Q 1 at the midpoint of a geodesic arc β labeled e i e i then we may simply identify the two arcs α and β to reduce the number of even boundary components of length two.
The other possibility is that γ reaches the midpoint of a geodesic arc β labeled e i+2 e i+2 . If this is the case attach two more polygons isometric to P along these edges and extend γ by one more edge. Now γ reaches a midpoint of a boundary geodesic arc labeled e i e i and we can proceed as before.
While the above operations might increase the total length of geodesic boundary arcs with odd labels or the total number of edges with even labels, they will reduce the number of geodesic boundary arcs of length two and even label. Let Q 2 denote the resulting surface. We point out that Q 2 is again a P-surface with locally convex boundary by Proposition 6.4.
(2) Identifying the remaining geodesic boundary arcs of length one with even labels. Every geodesic boundary arc of Q 2 with an even label has length one. Moreover Q 2 has the same number of edges labeled e i andē i for every even i by Proposition 6.3. Choose any bijection between these two sets and identify edges in pairs. Let Q 3 denote the resulting surface. Once again Proposition 6.4 implies that Q 3 is a P-surface with locally convex boundary.
(3) Attaching surfaces along totally geodesic boundary curves with odd labels to make all labels the same. Every boundary geodesic arc of Q 3 is locally convex and has an odd label. This implies that Q 3 has totally geodesic boundary and every boundary component maps to a power of one of the non-separating simple closed geodesic curves λ i with some orientation and i odd. Construct a family T 1,i of P-surfaces for every odd i ∈ {3, . . . , 2g − 1} by making use of Lemma 5.2. The boundary components of the surface T 1,i map only to powers of λ 1 and λ i with the same powers but with opposite orientations as in Q 3 . Moreover the total degree of the T 1,i 's is linearly bounded above in l(∂Q 3 ).
Let Q 4 be the surface obtained by attaching all of the T 1,i 's to the surface Q 3 in the obvious way. Proposition 6.4 shows that Q 4 is a P-surface.
(4) Capping off the remaining geodesic boundary arcs of a single odd label.
The boundary components of the P-surface Q 4 are all geodesic and map to λ r1 1 , . . . , λ rp 1 and λ l1 1 , . . . , λ lq 1 for some p and q in N. One direction of Lemma 5.1 implies that p+q is even and that p i=1 r i = q j=1 l j . The other direction allows us to construct a P-surface T of degree linearly bounded in l(Q 4 ) whose boundary components are the same as Q 4 but with opposite orientation. Let Q be the cover of S obtained by identifying the boundary of Q 4 with that of T in the obvious way.
To conclude observe that R embeds in the cover Q and that Area(Q\R) is bounded above linearly in l(∂Q 1 ) and hence in l(∂R), as required.

Proof of geometric flexible stability
Let S be a compact hyperbolic surface. Consider some * -cover p : X → S. We rely on Theorem 1.2 established in the previous section to complete the proof of Theorem 2.1 and therefore of our main result Theorem 1.1. Our strategy is to construct a cut graph Γ on X and then complete the complement X \ Γ to an unramified cover making use of Theorem 1.2.
Since X need not be a cover in the usual sense it might contain certain pathological closed curves that cannot exist on a cover of S. For example X might admit a simple closed curve γ such that a lift of p • γ to S admits self-intersections. It is clear that if X is ε-reparable then such a curve has to be eliminated. This motivates the following.
Proposition 7.1. Let Γ be a cut graph on X and C be any connected component of X \ Γ. Then C embeds in some cover of S as a subsurface with locally convex boundary.
Proof. Let C be a connected component of X \ Γ. It follows from the definition of cut graphs that C has no singular points of s(X) in its interior and that its boundary is locally convex. In particular C is non-positively curved and the map f = p |C : C → S is a local isometry. The result follows from Lemma 6.1.
We are now ready to prove that closed hyperbolic surfaces are flexibly geometrically stable.
Proof of Theorem 2.1. Let the constant ε > 0 be given. Take c > 0 to be sufficiently small so that c max{1, b S } ≤ ε. Let δ = δ(c) such that any * -cover X with β(X) < δArea(X) admits a c-cut graph, as provided by Theorem 4.2.
We claim that δ is as required in the definition of flexible geometric stability with respect to the given ε > 0. To see this consider a * -cover p : X → S with β(X) < δArea(X). We need to show that X is ε-reparable. Let Γ be a c-cut graph on X. In particular l(Γ) ≤ cArea(X) ≤ εArea(X).
Consider the complement C = X\Γ. Every connected component of C isometrically embeds into some cover of S according to Proposition 7.1. Relying on Theorem 1.2 we find a cover q : Q → S admitting an isometrically embedded copy of C on which the restriction of q agrees with the map p. Moreover where b S > 0 is the constant as in Theorem 1.2. This completes the verification that X is indeed ε-reparable.
The fact that Theorem 2.1 implies our main result Theorem 1.1 is contained in Proposition 2.4.

Appendix A. Voronoi and Delaunay on singular planes
We generalize some of the basic properties of the Voronoi tessellation and the Delaunay graph from the classical Euclidean and hyperbolic cases to the framework of a "hyperbolic plane with singularities". In particular we present the detailed proofs of Propositions 3.3, 3.4, 3.6 and 3.7 that were merely stated without a proof in Section 3.
Hyperbolic planes with singularties. Let S be a compact hyperbolic surface and p : X → S a fixed * -cover. Let q : X → X denote the universal cover of X equipped with the pullback length metric d X . The singular set of X is given by s( X) = q −1 (s(X)). The Voronoi cell A v at any vertex v ∈ s( X) and the Delaunay graph D were defined in Section 3.
We find it useful to introduce the following additional notation. Let m(v, u) denote the set of midpoints between any two given points v, u ∈ X. It is well-known [Bea12, §7.21] that the set of midpoints of any pair of points in the hyperbolic plane is the perpendicular bisector to the geodesic arc connecting these two points. Two vertices v 1 and v 2 of the Delaunay graph are connected by an edge if and only if there exists a mid- Lemma A.1. Let x 1 , x 2 , x 3 ∈ X be three distinct points. Then there is at most a single closed metric ball B ⊂ X withB ∩ s( X) = ∅ and {x 1 , x 2 , x 3 } ⊂ ∂B.
Proof. Assume that B ⊂ X is a closed ball as in the statement of the lemma. Let T ⊂ X be the geodesic triangle with vertices x 1 , x 2 , x 3 . The convexity of the ball B implies that T ⊂ B. Making use of Corollary 3.2 we may regard B as being isometrically embedded in the hyperbolic plane. The center x 0 of the ball B is determined by the triangle T . More precisely x 0 is the mutual intersection point of the three mid-point bisectors to the edges of T . The same reasoning shows that any other ball B ⊂ X as in the statement of the lemma necessarily has the same center as B and must therefore agree with B.
Proposition 3.3 says that every Voronoi cell is homeomorphic to a disc, is convex with a piecewise geodesic boundary and is determined by its Delaunay neighbors.
Proof of Proposition 3.3. Let v ∈ s( X) be a vertex of the Delaunay graph. The distance function d X (·, u) is continuous for every point u ∈ X. Therefore the Voronoi cell A v is closed. It is clear from the definition that A v ∩ s( X) = {v}. Since the singular set s( X) is co-bounded it follows that A v is compact. The same reasoning shows that there is a finite subset of vertices {u 1 , . . . , u n } ⊂ s( X) for some n ∈ N such that the Voronoi cell A v is given by If x ∈ ∂A v is a boundary point of A v then at least one of the above inequalities must be an equality. In other words ∂A v ⊂ A v ∩ n i=1 m(v, u i ). For every boundary point x ∈ ∂A v let F x ⊂ {u 1 , . . . , u n } be the non-empty subset defined such that u i ∈ F x if and only if x ∈ m(v, u i ). In other words every boundary point x ∈ ∂A v admits a closed ball B x ⊂ X centered at x with B x ∩ s( X) = ∅ and ∂B x ∩ s( X) = F x ∪ {v}. This relies on the fact that x ∈ A v and therefore d(x, v) ≤ d(x, u) for every vertex u ∈ s( X). Observe that every boundary point x ∈ ∂A v admits an open neighborhood U x so that Assume without loss of generality that U x is sufficiently small so that U x ⊂ B x . The ball B x is convex and can be regarded as being isometrically embedded in the hyperbolic plane by Corollary 3.2. We conclude that A v ∩ U x can be isometrically identified with a neighborhood of the point x inside a hyperbolic Voronoi cell.
Note that |F x | = 1 for all but finitely many boundary points x ∈ ∂A v according to Lemma A.1 . This condition is open in the sense that every x ∈ ∂A v with |F x | = 1 satisfies F x = F y for every y ∈ A v ∩ U x . Therefore ∂A v is locally a geodesic segment away from finitely many points where |F x | > 1. Moreover A v is locally convex at those points as well.
The above description of the boundary ∂A v in terms of the local hyperbolic picture with respect to the open cover U x shows that it is a disjoint union of simple closed curves. The Jordan curve theorem implies that A v is homeomorphic to a punctured closed disc. As A v is locally convex Lemma 3.1 implies that A v is in fact homeomorphic to a closed disc. Moreover A v is convex by the Cartan-Hadamard theorem. This concludes the proof of Items (1) and (2).
Note that a vertex u is adjacent to v in the Delaunay graph if and only if u = u i for some i ∈ {1, . . . , n} and there is a boundary point x ∈ ∂A v with F x = {u i }. In other words, the Delaunay neighbors of v correspond to the geodesic sides of the cell A v . In particular Item (3) of Proposition 3.3 follows.
Proposition 3.4 says that the interiors of distinct Voronoi cells are disjoint.
Proof of Proposition 3.4. Assume that the intersection A v,u = A v ∩ A u is nonempty. Every point x ∈ A v,u admits a closed metric ball B x ⊂ X centered at x withB x ∩ s( X) = ∅ and {v, u} ⊂ ∂B x . The ball B x is convex and so can be regarded as being isometrically embedded in the hyperbolic plane by Corollary 3.2.
Hyperbolic geometry implies that m(v, u) ∩ B x is a geodesic segment for every point x ∈ A v,u . Since A v,u ⊂ m(v, u) the convexity of A v,u implies that the intersection A v,u ∩B x is a geodesic segment as well. The compactness of A v,u allows us to extract a finite cover of such metric balls. It follows that A v,u is a finite union of geodesic segments. Since A v,u is convex it must be equal to a single point or a single geodesic segment. In particular A v,u has empty interior and soÅ v ∩Å u = ∅. Since both A v and A u are topological discs this implies ∂A v ∩Å u =Å v ∩ ∂A u = ∅. We conclude that A v,u ⊂ ∂A v ∩ ∂A u and so A v,u is a common geodesic side of the Voronoi cells A v and A u , as required.
The second statement of Proposition 3.4 concerning adjacency in the Delaunay graph follows from the last paragraph of the proof of Proposition 3.3.
Proposition 3.6 deals with the Delaunay graph and shows it is embedded.
Proof of Proposition 3.6. Let v 1 , u 1 and v 2 , u 2 be vertices of the Delaunay graph so that v i is adjacent to u i . Let x 1 and x 2 be points in X such that for i ∈ {1, 2} there is a closed metric ball B i ⊂ X centered at x i and withB i ∩ s( X) = ∅ and ∂B i ∩s( X) = {v i , u i }. Let γ i for i ∈ {1, 2} be the geodesic arc realizing the Delaunay edge between v i and u i and connecting these two points in X. In particular γ i ⊂ B i .
Assume towards contradiction that the two chords γ 1 and γ 2 intersect nontrivially along their interior. Recall that a pair of geodesics in a CAT(−1)-space admit at most a single intersection point. An examination of the resulting planar diagram shows that |∂B 1 ∩ ∂B 2 | > 2. Therefore Lemma A.1 applied with respect to any three distinct points of the intersection ∂B 1 ∩ ∂B 2 shows that the balls B 1 and B 2 must coincide. This is a contradiction.
The fact that the projection q(D) is embedded in X follows immediately from the above discussion combined with the π 1 (X)-invariance of the Delaunay graph.
Proposition 3.7 shows that the connected components of the complement of the Delaunay graph are locally convex.
Proof of Proposition 3.7. Assume towards contradiction that some connected component of X \ D is not locally convex. Equivalently, there is some Delaunay vertex v ∈ s( X) and some half-space H with v ∈ ∂H such that no Delaunay edge incident at v is contained in H. The proof of Lemma 3.8 and in particular the Claim contained in that proof shows that the Voronoi cell A v has infinite area 2 . This contradicts the fact that A v is compact.