Cubulating random quotients of hyperbolic cubulated groups

We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group.


Introduction
Gromov introduced the density model of random groups [Gro93,Chapter 9].Given a free group F , let S ℓ (F ) be the set of length ℓ words.For a density d ∈ (0, 1), choose a set R ⊂ S ℓ by selecting ⌊|S ℓ | d ⌋ elements from S ℓ , uniformly and independently.The associated random group at density d is the quotient F/⟨⟨R⟩⟩.Thus, for a rank r free group, a density d random quotient has the form ⟨a 1 , . . ., a r | g 1 , . . ., g k ⟩, where k ∼ (2r − 1) dℓ and the g i are independently chosen random words.
Gromov proved that with overwhelming probability as ℓ → ∞, random groups at density d are hyperbolic when d < 1 2 , and trivial or Z 2 when d > 1 2 .Ollivier proved that the same phase transition at density 1 2 occurs in quotients of any torsion-free hyperbolic group [Oll04].
1.1.Main result.Our main theorem is the following: Theorem 1.1.Let G = π 1 X, where X is a compact non-positively curved cube complex, and suppose that G is hyperbolic.Let b be the growth exponent of G with respect to the universal cover X.Let a be the maximal growth exponent of a stabilizer of an essential hyperplane of X.Let k ≤ e cℓ , where Then with overwhelming probability as ℓ → ∞, for any set of conjugacy classes [g 1 ], . . ., [g k ] with translation length |g i | ≤ ℓ, the group G/⟨⟨g 1 , . . ., g k ⟩⟩ is hyperbolic and is the fundamental group of a compact, non-positively curved cube complex.
In the above theorem, |g i | is the translation length of g i acting on the universal cover X; see Definition 2.1.The growth exponent of G with respect to X is a constant b > 0 such that the number of G-orbit points in an ℓ-ball in X is approximately e bℓ .See Definition 2.2 and Theorem 2.4 for a more precise characterization.The growth exponent of a hyperplane stabilizer is defined analogously.
A hyperplane U ⊂ X is essential if X is not contained in a finite neighborhood of U .A typical inessential hyperplane arises when X ∼ = Y × F , where F is finite and nontrivial.After subdividing, a compact non-positively curved cube complex X always deformation retracts to X ′ , where all hyperplanes of X ′ are essential.
The model of randomness employed in Theorem 1.1 is that we are sampling uniformly from the set of conjugacy classes whose translation length on X is at most ℓ.This departs from Gromov's density model in two small ways: we are permitting relators whose translation length is less than ℓ, and we are only counting one relator per conjugacy class.Ultimately, the exponential growth rate of balls in X is the same as the growth rate of spheres, and the growth of group elements is nearly the same as the growth of conjugacy classes (Theorem 2.4).Therefore, small variations in the model of randomness tend to have no effect on the conclusions one can reach about random quotients.See Ollivier ([Oll04, Sections 4-5] and [Oll05, Section I.2.c]) for a detailed and axiomatic discussion of this phenomenon.
The conclusion of Theorem 1.1 -that G = G/⟨⟨g 1 , . . ., g k ⟩⟩ is hyperbolic and cocompactly cubulated -has powerful consequences.A theorem of Agol [Ago13] implies G is virtually special, meaning G virtually embeds into a right-angled Artin group.Then, by a theorem of Haglund and Wise [HW10], all quasiconvex subgroups of G are separable.Furthermore, G is linear over Z [HW99, DJ00] and virtually surjects the free group F 2 [AM15].
1.2.Density and optimality of constants.In Theorem 1.1, the density of the random presentation G/⟨⟨g 1 , . . ., g k ⟩⟩ is c b .Thus all densities in the conclusion of the theorem are at most 1 41 , and might be lower, depending on the value of a.This is considerably lower than the densities appearing in the theorems surveyed at the start of the Introduction.The primary reason for needing the low density is that Theorem 1.1 is proved by establishing that random quotients of G satisfy the C ′ ( 1 20 ) cubical small cancellation condition.(See Definition 3.2 for the definition and Theorem 3.5 for the precise statement that C ′ ( 1 20 ) plus several mild hypotheses implies cubulation of the quotient.)Indeed, the probabilistic pigeonhole principle [Oll05,p. 31] implies that at any density larger than 1 40 there will almost surely be pieces that fellow-travel for more than 1 20 of their length.Thus our density hypotheses are nearly optimal for ensuring C ′ ( 1 20 ) small cancellation.
Strengthening Theorem 1.1 beyond density 1 40 would require one of two improvements.First, one could attempt to strengthen Theorem 3.5 and establish the cubulation of C ′ (α) quotients for some parameter α > 1 20 .Second, one could move away from small cancellation theory entirely, for instance by employing isoperimetric inequalities for van Kampen diagrams as in the work of Ollivier and Wise [OW11].This is also the approach employed in Ollivier's work on quotients of torsion-free hyperbolic groups [Oll04, Thm 3], which applies at all densities up to 1 2 but only ensures the hyperbolicity of the quotient.We remark that so long as G ≇ Z, the main theorem is non-vacuous, meaning a < b, because hyperplane stabilizers in G have strictly lower growth exponents than G itself.See [DFW19, Thm 1.3] and Theorem 2.3 below.Thus we may always pick c > 0 in Theorem 1.1.This leads to the following corollary: Corollary 1.2.Let X be a compact non-positively curved cube complex, such that π 1 X is hyperbolic and nonelementary.Then, at a sufficiently low density, generic quotients of π 1 X are cocompactly cubulated and hyperbolic.
For instance, if G is a surface group and the hyperplane stabilizers are cyclic (which occurs in the standard cubulations of G), we have a = 0, hence random quotients of G are cubulated and hyperbolic at density 1 41 .This is far lower than the density of 1 6 at which the corresponding conclusion is known for quotients of free groups.This supports our belief that the constants in Theorem 1.1 can be improved considerably, especially for surface groups.See Problem 7.1 and Problem 7.2.
1.3.Actions on other metric spaces.The growth of a group G is highly sensitive to the choice of metric space on which G acts.For instance, hyperbolic manifold groups in dimension n ≥ 3 admit canonical geometric actions on a hyperbolic space Υ = H n , but any cubulated group admits infinitely many distinct actions on non-isometric cube complexes.We may wish to study quotients of G by some number of relators that are sampled with respect to length in Υ rather than a cube complex X.Although lengths in X and Υ can be compared via an (equivariant) quasi-isometry, measuring lengths in the two spaces can lead to rather distinct samples of short words.
Nonetheless, there is an analogue of Theorem 1.1 for sampling words in G with respect to an action on Υ.To formulate the next theorem, we introduce the following nonstandard quantification of quasiisometries.A λ-quasiisometry is a coarsely surjective function f : X → Υ such that there exist positive constants λ 1 , λ 2 , ϵ with λ 1 λ 2 = λ, where every pair of points x, y ∈ X satisfies The product λ 1 λ 2 = λ remains unchanged if the metric on one of the spaces X or Υ is rescaled by a multiplicative constant.More generally, λ ≥ 1 has the following meaning.If G acts properly and cocompactly on both X and Υ, then these actions induce pseudo-metrics d 1 , d 2 on G itself.We call these pseudo-metrics roughly similar if they are related by a G-equivariant 1-quasiisometry.The space of rough similarity classes of pseudo-metrics on G is itself an interesting metric space D(G), studied topologically since the work of Furman [Fur02], and metrically since the work of Reyes [Rey23, Def 1.2].In Reyes's natural metric on D(G), the distance between [d 1 ] and [d 2 ] is precisely log λ for the optimal constant λ = λ 1 λ 2 in a G-equivariant quasiisometry X → Υ. See [Rey23,CR23a,BR].
We are interested in quotients of G, where the conjugacy classes of relators are drawn uniformly from among all elements of Υ-length less than ℓ.At sufficiently low density, depending on λ, these quotients are again hyperbolic and cubulated.
Theorem 1.3.Let G = π 1 X, where X is a compact non-positively curved cube complex, and suppose that G is hyperbolic.Suppose that G also acts properly and cocompactly on a geodesic metric space Υ, where every non-trivial element of G stabilizes a geodesic axis.Suppose that there is a G-equivariant λ-quasiisometry X → Υ.
Let b be the growth exponent of G with respect to Υ, and let a be the maximal growth exponent in Υ of a stabilizer of an essential hyperplane of X.Let k ≤ e cℓ , where Then with overwhelming probability as ℓ → ∞, for any set of conjugacy classes [g 1 ], . . ., [g k ] with translation length |g i | Υ ≤ ℓ, the group G = G/⟨⟨g 1 , . . ., g k ⟩⟩ is hyperbolic and is the fundamental group of a compact, non-positively curved cube complex.
As with Theorem 1.1, this result is non-vacuous, because hyperplane stabilizers in G have strictly lower exponential growth rates (with respect to Υ) than G itself.By Theorem 2.3 below, we have b > a, hence we may choose c > 0.
As a special case, suppose that a group G preserves a tiling of H 2 by regular rightangled pentagons.Let X be the square complex dual to the pentagonal tiling.Then every hyperplane is a line, hence hyperplane stabilizers are cyclic and have exponential growth rate a = 0.By a theorem of Huber [Hub59], generalized by Margulis [Mar69], the growth exponent of G with respect to H 2 is b = 1.Furthermore, in Proposition 6.1, we show that the optimal multiplicative constant in a λ-quasiisometry from X to H 2 is λ ≈ 1.5627.Consequently, Theorem 1.3 has the following corollary.
Corollary 1.4.Let S be a hyperbolic surface tiled by regular right-angled pentagons, and let G = π 1 S.For a number ℓ ≫ 0, let k ≤ e ℓ/63.51 , and let [g 1 ], . . ., [g k ] be conjugacy classes in G, chosen uniformly at random from among those of S-length at most ℓ.Then with overwhelming probability as ℓ → ∞, the group G = G/⟨⟨g 1 , . . ., g k ⟩⟩ is hyperbolic and is the fundamental group of a compact, non-positively curved cube complex.
In particular, Corollary 1.4 says that quotients of surface groups, with words sampled from a pentagonal hyperbolic metric, are cubulated and hyperbolic at density 1 64 .This is somewhat worse than sampling with respect to a cubical metric, where we obtain the same conclusion at density 1 41 ; see the discussion after Corollary 1.2.The multiplicative gap between these densities is essentially the constant λ.
Studying the effect of quasiisometry constants on density statements about quotients of G has led us to conjecture that there exist cubulations of a hyperbolic manifold group with quasiisometry constant λ arbitrarily close to 1. Equivalently, the points of D(G) corresponding to cocompact hyperbolic structures are limit points of metrics coming from cocompact cubulations.See Conjecture 7.8 and the subsequent discussion.This conjecture has been recently proved by Brody and Reyes [BR].
1.4.Overview.Section 2 reviews several results about the growth of a hyperbolic group, including the existence of a growth exponent and the statement that infinite-index quasiconvex subgroups have a lower growth exponent than the ambient group.Section 3 reviews the definitions of cubical small-cancellation theory and proves Theorem 3.5, a non-probabilistic cubulation criterion for C ′ ( 1 20 ) small-cancellation quotients of a cubulated group.This criterion is of independent interest, and has already been used in the work of Jankiewicz and Wise [JW22].
The probabilistic arguments supporting the proof of Theorem 1.1 are contained in Section 4. In that section, we control the sizes of pieces in a generic cubical presentation and show that below a certain density, a cubical presentation is C ′ ( 1 20 ), hence the quotient group is hyperbolic and cubulated.All of the geometric arguments in that section are coarse in nature, and it becomes natural to work with a certain generalization of pieces called loose pieces.The study of loose pieces also permits a translation between a G-action on a cube complex X and a G-action on a more general metric space Υ.We undertake this translation in Section 5, where we prove Theorem 1.3.Finally, in Section 6, we find the optimal multiplicative constants in a quasiisometry between a pentagonal tiling of H 2 and the dual cube complex, proving Proposition 6.1 and Corollary 1.4.
In Section 7, we collect some problems and questions motivated by these results.
1.5.Acklowledgements.We thank Eduardo Reyes and Sam Taylor for several enlightening conversations.We thank the referee for numerous minor corrections, and for suggesting a potential improvement to our results, as sketched in Remark 5.8 and Question 7.10.

Growth
This section collects several definitions and results about the growth of hyperbolic groups.None of the results recalled here are original.Definition 2.1 (Translation lengths).Let G be a group acting properly and cocompactly on a metric space Υ, and let g ∈ G be an infinite-order element.The translation length of g is defined to be |g| Υ = inf{d(x, gx) : x ∈ Υ}.The stable translation length of g is n , for an arbitrary x ∈ Υ.It is a standard property of isometries of metric spaces that the limit exists and is independent of x [BH99, page 230].
Observe that each of |g| Υ and g Υ only depends on the conjugacy class [g].By triangle inequalities, g Υ ≤ |g| Υ for every g.In addition, when Υ is hyperbolic, there is a constant C such that the following holds for every infinite-order g ∈ G: Finally, if g stabilizes a geodesic axis in Υ (as will typically be the case in our applications), we have g Υ = |g| Υ .Definition 2.2 (Growth).Let G be a finitely generated group acting properly and cocompactly on a metric space Υ.Fix a basepoint x ∈ Υ and a subset H ⊂ G.The growth function of H with respect to Υ is the function f H,Υ : N → N defined by Since G is a quotient of a finite-rank free group, and the action on Υ is proper, the growth function f H,Υ is no larger than exponential.Thus it makes sense to consider the logarithm of f .The growth exponent of H with respect to Υ is whenever the limit exists.We emphasize that the limit depends a great deal on Υ.However, triangle inequalities in Υ imply that ξ H (Υ) is independent of the basepoint.
Many results in the literature are expressed in terms of the growth rate λ H (Υ) = lim n f H,Υ (n) = e ξ H (Υ) instead of the growth exponent ξ H (Υ).However, the two notions carry the same information.
The following result was proved by Dahmani and the authors [DFW19, Thm 1.1], and independently Matsuzaki, Yabuki, and Jaerisch [MYJ20, Cor 2.8].See also [DFW19, Thm 1.3] for a statement that does not assume G is hyperbolic, but does assume Υ is a CAT(0) cube complex and H is a hyperplane stabilizer.
Theorem 2.3.Let G be a non-elementary hyperbolic group acting properly and cocompactly on a metric space Υ.Let H be a quasiconvex subgroup of infinite index.Then the growth exponents ξ H and ξ G exist, and The following theorem combines two results of Coornaert and Knieper [Coo93,CK02].Recall that a non-trivial element g ∈ G is primitive if g ̸ = h n for any n > 1. Primitive conjugacy classes are defined similarly.
Theorem 2.4.Let G be a nonelementary group acting properly and cocompactly on a δhyperbolic metric space Υ.Then, there exist positive constants A, B, b, n 0 , where b = ξ G (Υ), such that the following hold for n ≥ n 0 .
(1) The total number f G,Υ (n) of elements that translate a basepoint x ∈ Υ by distance at most n satisfies (2) Hence, for large n, the non-primitive conjugacy classes form a vanishingly small proportion of all conjugacy classes up to length n.

Cubical presentations and small-cancellation theory
This section reviews some definitions and results about cubical small-cancellation theory.Our primary reference is Wise [Wis21, Chapters 3-5].We also prove Theorem 3.5, a properness criterion that follows from [Wis21, Thm 5.44] but is easier to apply.
3.1.Cubical presentations and pieces.Definition 3.1 (Cubical presentation).A cubical presentation X * = ⟨X|Y 1 , . . ., Y m ⟩ consists of non-positively curved cube complexes X and Y i , and a set of local isometries Y i ↬ X.The cubical presentation X * corresponds to a topological space, also denoted X * , consisting of X with a cone on each Y i .Accordingly, we call each Y i a cone of X * .
See Figure 1 for an example.In this paper, it will always be the case that π 1 Y i ∼ = Z.However, this assumption is not present elsewhere in the literature.
For a hyperplane U of X, the carrier N ( U ) is the union of all closed cubes intersecting U .The systole ||X|| is the infimal length of an essential combinatorial closed path in X.In terms of Definition 2.1, ||X|| is the smallest translation length of a non-trivial element of π 1 X acting on X. Definition 3.2 (Pieces).Let X * = ⟨X|Y 1 , . . ., Y m ⟩ be a cubical presentation.A cone-piece of X * between Y i and Y j is a component of Y i ∩ Y j , for some choice of lifts of Y i and Y j to X, excluding the case where Given a constant α > 0, we say that X * satisfies the C ′ (α) small-cancellation condition if for every cone-piece or wall-piece P involving Y i , we have diam(P ) < α||Y i ||.
When pieces are small, the topological space X * satisfies a number of pleasant properties.
In a generalization of ordinary small-cancellation theory, small pieces guarantee the persistence of hypebolicity in X * : Lemma 3.4 ([Wis21, Lem 3.70 and Thm 4.7]).If π 1 X is hyperbolic and X * is compact and C ′ ( 1 14 ), then π 1 X * is hyperbolic.The main result of this section is the following theorem, which guarantees that π 1 X * acts properly on a CAT(0) cube complex.
Suppose that X is compact, and every Y i is compact and deformation retracts to a closed-geodesic.In In this example, as in Theorem 3.5, we have π 1 Y i ∼ = Z for each Y i , and the wallspace structure on Y i has two diametrically opposed hyperplanes in each wall.Unlike the setting of Theorem 3.5, π 1 X is not hyperbolic in this example.
addition, suppose that for every hyperplane U ⊂ Y i , the carrier N (U ) is embedded, the complement Y i ∖ U is contractible, and diam(N (U )) < 1 20 ||Y i ||.Then π 1 X * acts properly and cocompactly on a CAT(0) cube complex dual to a wallspace structure on X * satisfying the B(8) condition.
Moreover, if g ∈ π 1 X * ∖ {1} stabilizes a cell of the dual cube complex, then g is the image of an element g ∈ π 1 X such that a conjugate of some π 1 Y i lies in ⟨g⟩.In particular, if each π 1 Y i is maximal cyclic, then π 1 X * acts freely on X * .Theorem 3.5 will be proved with the aid of [Wis21, Thm 5.44 and Cor 5.45].Setting up the proof using those results requires a number of auxiliary definitions, beginning with the B(8) condition.
3.2.Wallspace small-cancellation conditions.We will construct a wallspace structure for X * .To do so, we will define a wallspace on each Y i , whose walls are equivalence classes of hyperplanes in Y i .This will generate a wallspace structure for X * whose walls are equivalence classes of hyperplanes generated by the equivalence relation fostered by the wallspace structures on the lifts Y i → X * .Definition 3.6 (B(6) and B(8) conditions).Suppose that 12 ) cubical presentation.A B(6) structure on X * consists of a wallspace structure on each Y i , which is preserved by its automorphisms, and satisfies a certain small-cancellation condition (see Figure 1 for an example).More precisely: (1) The collection of hyperplanes of each Y i is partitioned into classes satisfying the following conditions.No two hyperplanes in the same class cross or osculate; in particular, the carrier of each hyperplane u j embeds.The union U = ∪u j of the hyperplanes in a class is called a wall.Furthermore, for each wall U , there are halfspaces preserves the above wallspace structure on Y i .
(3) If P is a path in Y i that is the concatenation of at most 7 piece-paths and P starts and ends on the carrier N (U ) of a wall U , then P is path-homotopic into N (U ).The B(8) condition is defined by replacing (3) with the stronger condition: (3 ′ ) If P is a path that is the concatenation of at most 8 piece-paths and P starts and ends on the carrier N (U ) of a wall then P is path-homotopic into N (U ).
In the above, a piece-path in Y is a path in a piece of Y .Meanwhile, Aut(Y i → X) is the group of automorphisms ϕ : 3.3.The properness criterion.We now collect some terminology that will be needed to state and apply Theorem 3.8.In our usage, geodesics are globally distance realizing.By contrast, a closed-geodesic w → Y in a non-positively curved cube complex is a combinatorial immersion of a circle whose universal cover w lifts to a combinatorial geodesic w → Y in the universal cover of Y .We emphasize that (the image of) a closed-geodesic is not a geodesic in Y , because it is not distance realizing.
Let U be a hyperplane and v a 0-cube.We say that U is m-proximate to v if there is a path P = P 1 • • • P m such that each P i is either a single edge or a path in a piece, and v is the initial vertex of P 1 and U is dual to an edge in P m .A wall is m-proximate to v if has a hyperplane that is m-proximate to v.
A hyperplane u of a cone Y of X * is piecefully convex if the following holds: For any path ξρ → Y with endpoints on N (u), if ξ is a geodesic and ρ is either trivial or lies in a piece of Y containing an edge dual to u, then ξρ is path-homotopic in Y to a path µ → N (u).
(2) Each hyperplane of each cone Y i is piecefully convex.
(3) For each Y i , each infinite order element of Aut(Y i ) is cut by a wall of Y i .(4) Let κ → Y i be a geodesic with endpoints p, q.Let u 1 and u ′ 1 be distinct hyperplanes in the same wall of Y i .Suppose κ traverses a 1-cell dual to u 1 , and either u ′ 1 is 1-proximate to q or κ traverses a 1-cell dual to u ′ 1 .Then there is a wall U 2 in Y i that separates p, q but is not 2-proximate to p or q.Then π 1 X * acts with torsion stabilizers on the dual cube complex of the B(6) structure.
We refer to Figure 2 for a depiction of the notation in Hypothesis (4).We will now use Theorem 3.8 to prove Theorem 3.5.
Proof of Theorem 3.5.Observe that Definition 3.2 of the C ′ (α) condition involves a strict inequality.Since X * is a finite C ′ ( 1 20 ) presentation, we may choose a constant α < 1 20 such that X * is C ′ (α) and such that diam N (u) < α||Y i || for every hyperplane of every Y i .Fix such an α for the remainder of the proof.
Observe that cubical subdivision preserves all the hypotheses of the theorem, and the C ′ (α) condition in particular.We subdivide X a number of times, while retaining the original metric.That is: every original edge of X continues to have length 1, while every edge of the m th subdivision has length 2 −m .The reason for the iterated subdivision is that the length of an edge contributes additive error to several calculations below, and a large The scenario that arises in hypothesis (4) of Theorem 3.8.For the geodesic κ, we have to verify that neither p nor q is 2-proximate to value of m will make this additive error negligible.For instance, since α < 1 20 and ||Y i || ≥ 1, a large value of m ensures that Other inequalities with additive constants will follow similarly.
After the first subdivision of X, the hyperplanes of X and of Y i become 2-sided.By hypothesis, there is a closed-geodesic w i that generates π 1 Y i .Moreover, we may assume that Aut(Y i → X) stabilizes w i , for the following reason.Since Y i is compact, Y i is a quasi-line, hence Stab( Y i ) is virtually cyclic.Let g be the generator of the maximal cyclic subgroup in Stab( Y i ).Since we have subdivided X at least once, g stabilizes a geodesic axis v i ⊂ Y i , by [Hag23].Let v i = ⟨g⟩\ v i .Now, we may choose w i = ⟨g n ⟩\ v i , where g n generates π 1 (Y i ).
After the first subdivision, the closed-geodesic w i ⊂ Y i has an even number of edges.The hypothesis that every hyperplane of Y i has contractible complement implies that every hyperplane intersects some edge of w i .Our wallspace structure on each Y i will be defined by declaring each wall to consist of the pair of hyperplanes corresponding to a pair of antipodal edges of w i .(See Figure 1 for a partial illustration of such a pairing, where the closed-geodesics w i are presumed to run along the inner boundary of each Y i .) For the rest of the proof, we focus on one relator Y = Y i and its closed-geodesic w = w i .Before verifying the hypotheses of Theorem 3.8, we check an inequality that will be very useful in the sequel.
Let u, u ′ ⊂ Y be a pair of hyperplanes in the same wall of Y .This means that u, u ′ are dual to edges u, u′ ⊂ w.Recall that u has an embedded carrier such that diam(N (u)) < α||Y ||, and similarly for N (u ′ ).Now, let x ∈ N (u) and x ′ ∈ N (u ′ ) be arbitrary.Then 20 and m is chosen so that 2 −m is tiny.Next, we check the hypotheses of Theorem 3.8.In lieu of the B(6) condition of Definition 3.6, we will in fact verify the stronger B(8) condition.
To check Condition (1) of Definition 3.6, let u, u ′ be distinct hyperplanes in the same wall of Y .Then each of N (u) and N (u ′ ) is embedded by hypothesis.Furthemore, Equation (3.2) implies N (u) ∩ N (u ′ ) = ∅.Thus u ∪ u ′ partition Y into two halfspaces.Condition (2) of Definition 3.6 holds because our (revised) choice of w ensures that Aut(Y → X) stabilizes w, hence preserves the wallspace structure.
For Condition (3 ′ ), consider P = P 1 • • • P 8 , a concatenation of 8 piece-paths that starts on N (u) and ends on either N (u ′ ) or on N (u).We may assume that each P i is a geodesic.If Now, consider the case where P starts and ends on N (u).Since we have already checked that P is too short to reach N (u ′ ), it follows that P lifts to Y − (π 1 Y ) u ′ , each of whose components is convex.Thus P is path-homotopic into N (u).We have verified Condition (3 ′ ) of Definition 3.6.Hence X * satisfies the B(8) condition, and hypothesis 3.8.(1).
Hypothesis 3.8.(2),namely pieceful convexity of the hyperplanes of Y , follows from Remark 3.7.Indeed, let x ∈ N ( u) and x ′ ∈ gN ( u) be points on the carriers of two distinct lifts of u to Y .Then, by the same calculation as in Equations (3.1) and (3.2), we have d(x, x ′ ) > 8α||Y ||, which exceeds the diameter of a piece.
Hypothesis 3.8.(3)holds vacuously, since each Y is compact.We now verify Hypothesis 3.8.(4).Let κ → Y be a geodesic from p to q, as in that condition.We begin by deriving upper and lower bounds on |κ| = d(p, q).Let ṗ, q be vertices of w that are closest to p, q, respectively.Since a hyperplane carrier containing p cuts w and has diameter less than α||Y ||, we have d(p, ṗ) ≤ α||Y ||, and similarly d(q, q) ≤ α||Y ||.Since For the lower bound on |κ|, recall that there is a wall Otherwise, if q is 1-proximate to u ′ 1 , then d(q, u ′ 1 ) < α||Y ||, hence we obtain the weaker conclusion Now, let y be the midpoint of κ, and let u 2 be a hyperplane such that y ∈ N (u 2 ).Let u ′ 2 be the other hyperplane in the same wall U 2 .We claim that neither u 2 nor u ′ 2 is 2-proximate to p or q.To see this, we will check that d({p, q}, u 2 ) > 2α||Y ||, and similarly for u ′ 2 .For any x ∈ N (u 2 ), we have Here, the first inequality is the triangle inequality, the second inequality is the diameter bound on N (u 2 ), the third inequality is Equation (3.4), and the final inequality follows because α < 1 20 and 2 −m is tiny.Similarly, for any x ′ ∈ N (u ′ 2 ), we have Here, the first inequality is the triangle inequality, the second inequality is Equation (3.1), the third inequality is Equation (3.3), and the final inequality holds because α < 1 20 and 2 −m is tiny.Thus neither p nor q can be 2-proximate to u 2 or u ′ 2 .We have thus verified that Theorem 3.8 applies, hence π 1 X * acts with torsion stabilizers on the CAT(0) cube complex dual to the wallspace structure we have constructed.Since X is compact, this action is cocompact.
Finally, we check the conclusion about cell stabilizers in π 1 X * .We have already shown that cell stabilizers must be torsion.By [Wis21, Thm 4.2 and Rem 4.3], we know that when α < 1 20 , any torsion element in In particular, if π 1 Y i is maximal cyclic, then we obtain a contradiction, hence π 1 X * acts freely.□

Controlling pieces in generic quotients
This section contains the proof of the main theorem, Theorem 1.1.In the proof, we need to control the sizes of pieces in a generic cubical presentation.Wall-pieces are controlled in Proposition 4.7 and cone-pieces are controlled in a sequence of lemmas, culminating in Proposition 4.17.
Most of the work in this section occurs in the language of loose pieces, which represent a coarsening of the overlaps defined in Definition 3.2.We define loose pieces in Definition 4.4.
In Lemma 4.5, we show that every wall-piece or cone-piece in the sense of Definition 3.2 gives rise to a corresponding loose piece.
Via loose pieces, the proofs of Propositions 4.7 and 4.17 readily generalize to the context of non-cubical hyperbolic metric spaces in Section 5. 4.1.Loose pieces and convex hulls of quasi-axes.Let Υ be a metric space.For any subset s ⊂ Υ and J > 0, the notation N J (s) denotes the closed J-neighborhood of s.
Definition 4.1 (Quasi-axes).Let w, w ′ be bi-infinite geodesics in a metric space Υ.We declare them to be equivalent, and write w ≈ w ′ , whenever w ⊂ N r ( w ′ ) for some r ≥ 0. In the main case of interest, when Υ is δ-hyperbolic, we can use a uniform value r = 2δ, and deduce that w ≈ w ′ if and only if Let g be a hyperbolic isometry of Υ.A quasi-axis for g is a bi-infinite geodesic w ⊂ Υ such that g w ≈ w.When Υ is δ-hyperbolic, any geodesic connecting the fixed points of g on ∂Υ is a quasi-axis for g.Lemma 4.2.Let X be a δ-hyperbolic, finite dimensional CAT(0) cube complex.Then there is a uniform constant K = K( X) with the following property.For every bi-infinite geodesic w, the convex hull of w satisfies Proof.Consider a vertex p ∈ X that lies far from w.Let s be a shortest CAT(0) (noncombinatorial) geodesic segment from p to w, oriented outward from p. Sageev and Wise have observed [SW15, Remark 3.2] that the first cube met by s contains a midcube of a hyperplane H that intersects s at an angle bounded away from 0. The angle bound depends only on the dimension of the cube, hence can be taken uniformly over X.
We claim that when s is sufficiently long, H ∩ w = ∅.In other words, H separates p from w, hence p / ∈ hull( w).The claim holds because an intersection H ∩ w would determine a CAT(0) geodesic triangle with a long base along s and two large angles along s.Such a triangle cannot be δ-thin.□ The following construction relates cubical presentations to quasi-axes.Given G = π 1 X and g ∈ G − {1}, choose any quasi-axis w for g.Then, construct a convex subcomplex Y = hull( w) ⊂ X, whose quotient Y = ⟨g⟩\ Y admits a local isometry into X.By Lemma 4.2, Y lies in a uniform neighborhood of w, hence its quotient Y is compact.We say that Y is a quasi-circle defined by g.
If g stabilizes an axis w, as will be the case following Convention 4.3, then the quasi-circle Y deformation retracts to the closed-geodesic w = ⟨g⟩\ w.Now, we adopt the following convention.
Convention 4.3.From now until the start of the proof of Theorem 1.1, the symbol X denotes a compact non-positively curved cube complex, such that X is δ-hyperbolic for some δ.We assume that G = π 1 X is nonelementary.We assume that X has been subdivided at least once, so that every g ̸ = 1 stabilizes a geodesic axis in X [Hag23].
Finally, we assume that every hyperplane of X is essential.This assumption is harmless, because every non-positively curved cube complex X ′ contains a convex essential core, whose hyperplanes are essential.See, for instance, Caprace and Sageev [CS11, Proposition 3.5].
Our arguments in this section will employ the following, looser analogue of Definition 3.2.Definition 4.4 (Loose pieces).Let X be as in Convention 4.3.Fix a constant J ∈ N. We define J-loose pieces first in X, then in X.
Consider bi-infinite geodesics w, w ′ that do not share an endpoint in ∂ X.Then a J-loose cone-piece between w and w ′ is the maximal geodesic segment s ⊂ w whose endpoints are contained in N J ( w ′ ).The companion of s is the maximal segment s ′ ⊂ w ′ whose endpoints are at distance J from the corresponding endpoints of s.Now, let w, w ′ be closed-geodesics in X.If w ̸ = w ′ , a J-loose cone-piece between w and w ′ is the projection of a J-loose cone-piece between preimages w and w ′ .A J-loose cone-piece between w and itself arises in the above scenario where w ′ = h w for some h / ∈ Stab G ( w).In a similar manner, a J-loose wall-piece in w is a maximal geodesic segment s ⊂ w whose endpoints are contained in N J (N ( U )) for a hyperplane U .A J-loose wall-piece in a closed-geodesic w is the projection of a J-loose wall-piece in w.
The next lemma shows that there is a uniform value J = J( X) such that every piece in a cubical presentation X * must correspond to a J-loose piece in X.
Lemma 4.5.Consider a cubical presentation X * = ⟨X | Y 1 , . . ., Y m ⟩.Suppose that X is hyperbolic and that every Y i is a quasi-circle.Let w i be a bi-infinite geodesic in Y i .Then there is a constant J = J( X) such that the following hold for every d ≥ 0: (1) Every diameter ≥ d cone-piece of X * between Y i and Y j determines a J-loose cone-piece between w i and w j , of diameter ≥ d.
(2) Every diameter ≥ d wall-piece of X * in Y i determines a J-loose wall-piece in w i , also of diameter ≥ d.
(3) Every diameter ≥ d intersection between Y i and a hyperplane carrier N (U ) determines a J-loose wall-piece in w i , of diameter ≥ d.
Furthermore, we may take J ≤ 3K, where K is the constant of Lemma 4.2.
Proof.First, consider cone-pieces.Let p, q ∈ Y i ∩ Y j be points such that d(p, q) ≥ d.By Lemma 4.2, there are points p i , q i ∈ w i that are K-close to p and q, respectively.Applying Lemma 4.2 to the quasi-circle Y j shows that p i , q i are 2K-close to w j .Since w i is a geodesic, we have d(p i , q i ) ≥ d − 2K.Extending the segment [p i , q i ] by K in each direction, we obtain a geodesic segment [p ′ i , q ′ i ] ⊂ w i whose length is at least d and whose endpoints lie in N 3K ( w j ).Thus [p i , q i ] is contained in a 3K-loose cone-piece between w i and w j .
The proof for wall-pieces and intersections with hyperplanes is identical.In this case, we may take J = 2K.□ Remark 4.6.Besides the looseness constant J, there is one important respect in which Definition 4.4 is more general than Definition 3.2.Whereas the definition of a wall-piece in Y i requires the hyperplane U to be disjoint from Y i , the definition of a J-loose wall-piece in w i has no analogous restriction.Consequently, the J-loose wall-pieces in Lemma 4.5.(3)need not come from wall-pieces.When we apply Theorem 3.5, we will care about all intersections between Y i and hyperplane carriers in X, and all such intersections will be controlled using J-loose wall-pieces.
4.2.Wall-pieces.Our next goal is to give a criterion on the growth of the number of relators ensuring that with overwhelming probability, the diameter of wall-pieces in a cubical presentation is bounded by α||Y i ||.We formulate the criterion in terms of loose pieces, for greater applicability in Theorem 3.5 and in the generalized setting of Section 5.
Recall that by Convention 4.3, we are assuming that G is nonelementary and the hyperplanes of X are essential.By Theorem 2.4, the growth function of G acting on X is bounded by constants times e bℓ , where b > 0 is the growth exponent.Let a the largest growth exponent among the carriers of the finitely many orbits of hyperplanes of X.Then a is also the largest growth exponent of any of the J-neighborhoods of the hyperplanes, for an arbitrary J. Recall that a < b by Theorem 2.3.
For each ℓ > 0, let G(ℓ) denote the set of nontrivial conjugacy classes in π 1 X of length at most ℓ.For the duration of this section, we will be sampling randomly from G(ℓ), for large ℓ.
Proposition 4.7.Let X be as in Convention 4.3.For each non-trivial conjugacy class π 1 X, choose a representative closed-geodesic.Let a < b be growth exponents as above, let α ∈ (0, 1), and fix a positive number c < α(b − a).Then, for every J ∈ N, there is a constant C = C(J, X), with the following property.
For k ≤ e cℓ , choose conjugacy classes Then, for all ℓ ≫ 0, the probability that some w i contains a J-loose wall-piece of diameter at least α||w i || is bounded above by Consequently, with overwhelming probability as ℓ → ∞, all J-loose wall-pieces in w i have diameter less than α|g i |.
Proof.Let Π(α, ℓ) denote the proportion of representative closed-geodesics w of length at most ℓ such that w contains a J-loose wall piece of diameter at least α||w||.That is, Π(α, ℓ) denotes the proportion of all w of length at most ℓ such that w ∩ N J (N ( U )) contains the endpoints of a segment of length at least α||w|| for some hyperplane U .Our goal is to estimate Π(α, ℓ).
For each n ≤ ℓ, let U(n, α) be the set of (conjugacy representatives of) closed-geodesics w of length exactly n that contain a J-loose wall-piece s with |s| ≥ αn.Then Each element w ∈ U(n, α) can be cyclically permuted to be of the form s • y, where s lies in N J (N ( U )) for some hyperplane U , and |s| ≥ αn.By Theorem 2.4.(1) and Remark 2.5, the number of such paths s, up to homotopy in X, is bounded above by B ′ e a|s| , for some constant B ′ depending on J. Similarly, the number of paths y is bounded above by B ′′ e b|y| , for some constant B ′′ .
Let B = B ′ B ′′ .Since there are at most n distinct cyclic permutations of w, the number of elements in U(n, α) is Summing over all lengths up to ℓ, we obtain: .
Among all choices (w 1 , w 2 , . . ., w k ) ∈ G(ℓ) k , the proportion of k-tuples where some w i ∈ U(n, α) is bounded above by kΠ(α, ℓ).Since k ≤ e cℓ , Equation (4.2) says that the total probability of a large piece is bounded by completes the proof of the estimate.In particular, since c+(a−b)α < 0 by hypothesis, we have kΠ(α, ℓ) → 0 as ℓ → ∞. □ 4.3.Cone-pieces between two closed-geodesics.Our next goal is to control conepieces whose diameter is bounded below by a constant d.The gist of the following sequence of lemmas is that the probability of a J-loose cone-piece of diameter at least d declines exponentially with d.See Proposition 4.14.We will need to employ separate arguments for pieces between closed-geodesics w and w ′ chosen separately, and pieces between a closed-geodesic w and itself.In the case of a piece between w and itself, we will further need to consider overlaps, defined as follows.
Definition 4.8 (Orientation and overlaps).Let w → X be a closed-geodesic.Choose distinct preimages w and h w of w and a constant J ∈ N. Suppose s = [p, q] ⊂ w is a J-loose piece between w and h w, such that |s| > 2J.Let s ′ = [p ′ , q ′ ] ⊂ h w be the companion of s, so that d(p, p ′ ) = J = d(q, q ′ ).
Suppose that w is oriented from p to q, and transfer this orientation to h w via h.Since |s| = d(p, q) > 2J, the endpoints p ′ , q ′ of the companion s ′ cannot coincide.We say that h preserves orientation on s if p ′ comes before q ′ in the orientation on h w, and reverses orientation on s otherwise.See Figure 3.
The h-overlap in s is the interval u = s ∩ h −1 s ′ .
The main steps of the proof of Proposition 4.14 can be organized as follows: The terminology and setup of Definition 4.8.The J-loose piece s = [p, q] is shown in yellow, and the companion s ′ = [p ′ , q ′ ] in orange.In this example, h reverses orientation on the piece s, and • In Lemma 4.10, we show that any orientation-reversing piece between w and itself has small overlap, universally bounded by a constant R.This is an unconditional (non-probabilistic) statement.• In Lemma 4.11, we show that pieces between separately sampled closed-geodesics w and w ′ are exponentially rare.Similarly, pieces between w and itself with small overlap (of size ≤ R) are exponentially rare.• In Lemma 4.13, we show that pieces between w and itself with big overlap (of size ≥ R) are also exponentially rare.This allows us to conclude that each type of piece of diameter ≥ d is exponentially rare.
We will need the following standard fact about thin quadrilaterals.
Lemma 4.9.Suppose that X is δ-hyperbolic.Let Q be a geodesic quadrilateral with corners p, q, q ′ , p ′ , such that d(p, p ′ ) ≤ J and d(q, q ′ ) ≤ J. Then every point x ∈ [p, q] lies within distance J = J + 2δ of some point of [p ′ , q ′ ].
Proof.This is standard.Since triangles in X are δ-thin, then quadrilaterals are 2δ-thin.Thus x ∈ [p, q] is 2δ-close to some point Lemma 4.10.Suppose X is compact and X is δ-hyperbolic.For every J ∈ N, there is a constant R = R(J, δ) ≥ 10J such that the following holds.Suppose that s ⊂ w is a J-loose cone-piece between w and h w, where h reverses orientation on s.Then the h-overlap u ⊂ s has diameter |u| ≤ R.
Proof.In the notation of Definition 4.8, the overlap is u Since h is orientation-reversing, the endpoints of u are x ∈ {p, h −1 q ′ } and y ∈ {q, h −1 p ′ }.We begin by showing there is a constant L = L(J, δ) such that (4.3) d(x, hy) ≤ L and d(y, hx) ≤ L.
The above quasigeodesics must uniformly fellow-travel v.As a consequence, there is a uniform radius r, depending only on J and δ, such that ⟨h⟩(x) ⊂ N r ( v).Lemma 4.11.Suppose that every non-trivial element of π 1 X stabilizes a geodesic in a δ-hyperbolic space X.For each non-trivial conjugacy class in π 1 X, choose a closed-geodesic in X that represents it.
Fix constants J, R ∈ N.Then, for all d > 2J and all ℓ sufficiently large, the following holds: (1) Among all pairs of conjugacy classes [g], [g ′ ] of length at most ℓ, the proportion whose representative closed-geodesics have a J-loose cone piece of diameter ≥ d is less than M ℓ 2 e −bd , for a constant M = M (X, J).
(2) Among all conjugacy classes [g] of length at most ℓ, the proportion whose representative closed-geodesic has a J-loose cone piece with itself, with diameter ≥ d and overlap of length ≤ R, is less than M 0 ℓ 2 e −bd , for a constant M 0 = M 0 (X, J, R).
Proof.Before beginning the probabilistic portion of the proof, we make a reduction, introduce notation, and name some constants.Let D be the diameter of X.Let J = J + 2δ, and let V be the maximal number of vertices in a ball of radius (D + J + J) in X.Let W be the maximal number of vertices in a ball of radius n 0 , where n 0 is the threshold constant of Theorem 2.4.Fix a basepoint x 0 ∈ X.
Suppose that [g], [g ′ ] are conjugacy classes represented by closed-geodesics w, w ′ that have a J-loose cone-piece.(For now, we do not place any constraint on the lengths of w, w ′ , or the piece.)By Definition 4.4, the piece in w is the image of a geodesic segment s ⊂ w ⊂ X, which is a J-loose cone piece between w and w ′ .By choosing the preimage s appropriately, we ensure the additional property that the initial point p ∈ s lies within radius D of the basepoint x 0 ∈ X.Let s ′ ⊂ w ′ be the companion of s.Then the initial point of s ′ is a point p ′ that lies J-close to p ∈ S, hence within distance D + J of the basepoint x 0 .Furthermore, by Lemma 4.9, every point of s ′ lies within distance J = J + 2δ of some point of s.
Since conjugacy classes in π 1 X correspond to free homotopy classes in X, every geodesic segment in w ′ of length |w ′ | is a fundamental domain for w ′ .It will be convenient to choose a fundamental domain v ′ ⊂ w ′ that begins at p ′ , such that the initial segment of v ′ coincides with the initial segment of s ′ .Since the orientation of s ′ (determined by the condition that the initial point p ′ ∈ s ′ is J-close to p ∈ s) may or may not coincide with the orientation of w ′ , we conclude that the segment v ′ chosen as above is a fundamental domain for (w ′ ) ±1 .Now, we proceed to the proof of (1).Fix a conjugacy class [g] and its representative closed-geodesic w.We will bound the number of conjugacy classes [g ′ ], with representative closed-geodesic w ′ , such that w has a J-loose cone-piece with w ′ , of diameter at least d.
As in the above notation, let s ⊂ w be a preimage of the piece in w that begins within distance D of the basepoint x 0 .As in the opening paragraph of the proof, let s ′ ⊂ w ′ be the corresponding sub-segment of w ′ , so that the initial point p ∈ s is J-close to the initial or terminal point p ′ ∈ s ′ .
As a warm-up case, suppose that |s ′ | ≥ |w ′ | − n 0 , where n 0 is the threshold constant in Theorem 2.4.As above, (w ′ ) ±1 has a fundamental domain v ′ ⊂ w ′ whose initial segment coincides with the initial segment of s ′ .More precisely, we have a fundamental domain and d(q ′ , r ′ ) ≤ n 0 .Then, by construction, we have d(p, p ′ ) = J.Furthermore, by Lemma 4.9, we have d(q ′ , q) ≤ J for some point q ∈ s.Observe that |d(p, q) − d(p ′ , q ′ )| ≤ J + J.
By the definition of V , and the choice of s so that its start lies close to the basepoint, there are at most V choices for where p ′ can lie.Then, once n = d(p, q) is chosen, the point q ∈ s is determined, hence there are at most V choices for q ′ ∈ s ′ .Since d(q ′ , r ′ ) ≤ n 0 , it follows that once q ′ is chosen, there are at most W choices for r ′ .Consequently, for every n, there are at most V 2 W choices for the segment v ′ , hence at most 2V 2 W choices for [g ′ ], where the factor of 2 accounts for the orientation of g ′ .Recalling that n ≤ ℓ + J + J, the total number of possibilities for [g ′ ] is at most In particular, the bound grows linearly with ℓ in the (non-generic) warm-up case.
Having finished the warm-up case, assume that |s ′ | < |w ′ | − n 0 .Then, as above, (w ′ ) ±1 has a fundamental domain v ′ ⊂ w ′ whose initial segment coincides with that of s ′ .Since We first bound the number of possibilities for s ′ .Since the endpoints of s ′ are J-close to those of s, for every choice of s there are at most 2V 2 choices for where s ′ begins and ends.(The factor of 2 comes from the choice of direction of fellow-traveling.)Turning attention to y ′ , observe that the endpoint of s ′ is the starting point of y ′ .Since |y ′ | ≥ n 0 , Theorem 2.4.(1) and Remark 2.5 imply that there are at most Be b|y ′ | choices of where y ′ ends.Thus, for every choice of s, the number of choices for (s ′ y ′ ) ±1 up to path-homotopy is at most Recall that the companion s ′ is determined by the J-loose cone-piece s ⊂ w.There are |w| ≤ ℓ possible choices of where the projection of s begins in the closed-geodesic w.There are also choices for the length |s|, constrained by the inequalities d ≤ |s| < |w ′ | ≤ ℓ.Summing over these possible choices, we conclude that for every conjugacy class [g], the number of possibilities for [g ′ ] such that the J-loose cone-piece s has diameter at least d, is bounded above by (4.5) By increasing the constant B if needed, we may ensure that the above exponential bound subsumes the linear bound in the non-generic warm-up case.
Next, recall that G(ℓ) denotes the set of nontrivial conjugacy classes of translation length at most ℓ.By Theorem 2.4.(2), the number of such conjugacy classes satisfies Dividing the previous two equations produces a bound on the fraction of conjugacy classes [g ′ ] whose representative closed-geodesic w ′ has a J-loose cone-piece of diameter at least d with w.This fraction is at most Here, M = 2V 2 Be 2bJ A(1−e −b ) is a constant that depends only on X and J.The proof of Conclusion (2) is very similar.Suppose that s ⊂ w is a J-loose cone piece between w and h w, of diameter |s| ≥ d > 2J.Since d is large, Definition 4.8 applies.Let s ′ ⊂ h w be the companion of s, and let u = s ∩ h −1 (s ′ ) be the overlap.By hypothesis, we have |u| ≤ R. Let (v ′ ) ±1 = s ′ ∖ h(u) = s ′ ∖ h(s) be the portion of s ′ that is not in the image of the overlap.Then s and v ′ project to disjoint portions of the closed-geodesic w ⊂ X, and the sign ±1 is chosen so that s and v ′ are oriented consistently along w.Thus there is a fundamental domain for w of the form y 1 v ′ y 2 s.
Observe that |y 1 v ′ y 2 | ≤ ℓ−|s|, and the subsegment v ′ of length |v ′ | ≥ |s ′ |−R ≥ |s|−2J −R is entirely determined by s and a bounded amount of extra data.Thus, as in Equation (4.4), we compute that for every choice of s, the number of choices for y 1 v ′ y 2 up to path-homotopy is at most 2ℓ V 2 Be b(ℓ−2|s|+2J+R) .Compared to (4.4), the extra factor of ℓ comes from the choice of where in the fundamental domain the subword v ′ occurs, and the extra constant R in the exponent comes because we have removed the overlap from s ′ .Since there are at most Be b|s| choices for s, we conclude that the number of possibilities for w is bounded above by The remainder of the proof of (1), comparing the above upper bound to the lower bound G(ℓ) ≥ (A/ℓ)e bℓ , goes through verbatim.We conclude that the fraction of conjugacy classes [g] whose representative closed-geodesic has a J-loose cone-piece with itself, with diameter ≥ d and overlap of length ≤ R, is bounded by A .
Setting M 0 = 2V 2 B 2 e b(2J+R) /A completes the proof.□ Lemma 4.12.Suppose X is compact and X is δ-hyperbolic.Choose a non-trivial element h ∈ π 1 X.Suppose that yx is a path from p ∈ X to hp ∈ X. Suppose that each of x and y is a geodesic, and that |x| ≤ J for some J ≥ 0. Define L to be the maximal length of a terminal segment of y that is a 2δ fellow-traveler with an initial segment of hy.Let (yx) ∞ denote the bi-infinite path ∞ is an η-quasigeodesic, with η depending only on J, δ, and L.
Proof.If |y| is small, there are only finitely many possibilities for yx.The bi-infinite path (yx) ∞ lies at bounded distance from some quasi-axis of h.So, take the worst case scenario for the quasigeodesic constant η.
If |y| is large, we have large geodesic subpaths of (yx)h(y).By hypothesis, we have a bound L on the length of backtracking in (yx)h(y).Now, apply a "local to global" principle for quasi-geodesics.See e.g.Cannon [Can84,Thm 4].□ We can now use Lemmas 4.10, 4.11, and 4.12 to show that large J-loose pieces between a closed-geodesic and itself are exponentially rare.Lemma 4.13.Suppose X is compact and X is hyperbolic.Fix a constant J ∈ N. Then there is a constant M = M (X, J) such that for all sufficiently large ℓ and for all d ∈ [2J + 1, ℓ/3], the following holds.Among all conjugacy classes [g] of length at most ℓ, the proportion whose representative closed-geodesic has a J-loose cone piece with itself, with diameter ≥ d, is less than M ℓ 2 e −bd .
Proof.Consider a closed-geodesic w representing the conjugacy class [g].As a warm-up, we dismiss the (non-generic) possibility where |w| ≤ 2d.Since d ≤ ℓ/3, it follows that |w| ≤ 2ℓ/3.By Theorem 2.4.(2), the total number of conjugacy classes of length at most 2ℓ/3 is at most Be 2ℓ/3 , whereas the total number of length at most ℓ is at least Ae bℓ /ℓ.Thus, the probability that |w| is at most 2/3 of the allowed length is Thus the conclusion of the lemma holds for M 1 = B/A.Note that this warm-up case did not use any hypotheses about pieces between w and itself.Now, suppose that a closed-geodesic w representing the conjugacy class [g] has a J-loose cone-piece with itself.Following Definition 4.4, let s ⊂ w be a preimage of the piece in w, where the endpoints of s lie at distance J from h w.Following Definition 4.8, let s ′ ⊂ h w be the companion of s, and let u = s ∩ h −1 s ′ be the h-overlap in s.
Let R = R(J, δ) ≥ 10J be the constant of Lemma 4.10.By Lemma 4.11.(2),pieces with an overlap of diameter |u| ≤ R occur with probability at most M 2 ℓ 2 e −bd , where M 2 = M 2 (X, J) = M 0 (X, J, R(J, δ)) in the notation of Lemma 4.11.Thus, we may suppose that |u| > R. By Lemma 4.10, h preserves the orientation on s.We now consider two cases.
Case 1: |s| ≤ 1 2 |w|.Without loss of generality, suppose that the overlap u ⊂ s occurs at the beginning of u.Then there is a sub-geodesic y 1 y 2 y 3 ⊂ w, where s = y 2 y 3 and s ′ = h(y 1 y 2 ), so that y 2 = s ∩ h −1 s ′ is the h-overlap.Then |y 2 | ≥ R ≥ 10J, where the second inequality comes from Lemma 4.10.Consequently, . Bottom: y 1 y 2 is a geodesic, and hy 1 fellow-travels y 2 .Hence there is a bounded amount of fellow-travelling between y 1 and hy 1 .This implies (y 1 x 1 ) ∞ is a quasigeodesic, and similarly for (y 3 x 3 ) ∞ .The two quasigeodesics must δ ′ -fellowtravel, hence there is a geodesic of length δ ′ from the endpoint of y 3 to some point on (y 1 x 1 ) ∞ .
and we can conclude that |y 1 y 2 y 3 | < |w|.Thus we may choose a fundamental domain for w of the form y 1 y 2 y 3 z.Let x 1 denote a geodesic from the endpoint of y 1 to the start of h(y 1 ), and let x 3 denote a geodesic from the endpoint of y 3 to the start of h(y 3 ).Then |x 1 |, |x 3 | ≤ J. Since y 1 y 2 is a geodesic and h(y 1 ) fellow-travels with y 2 , we see that in the path y 1 x 1 h(y 1 ) there is a uniform upper bound on the amount of (2δ)-fellow-traveling between the terminal subpath of y 1 and the initial subpath of h(y 1 ).Thus, in the notation of Lemma 4.12, we conclude that (x 1 y 1 ) ∞ is an η-quasigeodesic for some η = η(δ, J) > 0. Likewise, (y 3 x 3 ) ∞ is an η-quasigeodesic.These quasigeodesics fellow-travel, since they are periodically joined by translates of y 2 .See Figure 4.
Let δ ′ = δ ′ (δ, J) be a uniform constant such that η(δ, J)-quasigeodesic quadrilaterals in X must be δ ′ -thin.Then it follows that (x 1 y 1 ) ∞ and (y 3 x 3 ) ∞ are δ ′ -fellow-travelers.Since the endpoint of y 3 is δ ′ -close to (x 1 y 1 ) ∞ , we have an η-quasigeodesic triangle with sides consisting of a sub-path of (x 1 y 1 ) ∞ , a copy of (y 2 y 3 ), and a geodesic segment of length at most δ ′ (shown dashed in Figure 4).Since the dashed segment is δ ′ -short, all of y 2 y 3 is 2δ ′ -close to (x 1 y 1 ) ∞ .Since |x 1 | ≤ J, all of y 2 y 3 is determined by y 1 and a universally bounded amount of data.Since |y 2 y 3 | ≥ d, the same argument as in Lemma 4.11.(2),shows that the probability of a J-loose cone-piece with this configuration is at most M 3 ℓ 2 e −bd , for a constant M 3 = M 3 (X, J).
Case 2: |s| > 1 2 |w|.By the warm-up argument at the beginning of the proof, we may assume d ≤ 1 2 |w| < |s|.As in Case 1, we may assume without loss of generality that the overlap u occurs at the beginning of s.Let s be the terminal sub-geodesic of s, of length |s| = d.By Lemma 4.9, there is a sub-geodesic s′ ⊂ s ′ whose endpoints are J-close to those of s, for J = J + 2δ.Let ů = s ∩ h −1 (s ′ ).We think of s as a sub-piece and ů as a sub-overlap.
If |ů| ≤ R = R(J, δ), observe that s and v′ = s′ ∖ h(ů) project to disjoint portions of the closed-geodesic w, and that v′ is determined by s and a universally bounded amount of extra data.Thus, by Lemma 4.11.(2), the probability of a J-loose cone-piece containing this configuration is at most M 4 ℓ 2 e −bd , where M 4 = M 4 (X, J) = M 0 (X, J, R(J, δ)) in the notation of Lemma 4.11.
If |ů| ≥ R = R(J, δ), we employ the argument of Case 1, with J in place of J, to show that the probability of a J-loose cone-piece with this configuration is at most M 5 ℓ 2 e −bd , for a constant M 5 = M 5 (X, J).
To complete the proof, recall the constants M 1 , . . ., M 5 defined above, so that the probability of a J-loose cone-piece of each type is bounded by M j ℓ 2 e −bd for the appropriate M j .Setting M = max j M j completes the proof.□ Proposition 4.14.Suppose X is compact and X is hyperbolic.Fix constants J ∈ N and ϕ ∈ (0, 1) and C > 2/b(1 − ϕ).Then, for all sufficiently large ℓ and for all d ∈ [C log ℓ, ℓ/3], the following holds.Suppose that conjugacy classes [g], [g ′ ] are chosen at random from among those of length at most ℓ.Then there is an e −bdϕ upper bound on the probability that there is a J-loose piece of diameter at least d between the representative closed-geodesics.
Similarly, for a randomly chosen conjugacy class [g] of length at most ℓ, there is an e −bdϕ upper bound on the probability that there is a J-loose piece of diameter at least d between the representative closed-geodesic and itself.
Proof.Let M = M (X, J) be the larger of the two constants in Lemma 4.11.(1) and Lemma 4.13.Combining the two lemmas, we see that a J-loose cone-piece of diameter at least d in the closed-geodesic representing [g] occurs with probability at most M ℓ 2 e −bd .Now, let C > 2 b(1−ϕ) be as in the statement of the lemma.Then, choosing ℓ large ensures that the additive difference C log ℓ − 2 log ℓ b(1−ϕ) is as large as we like.In particular, for ℓ ≫ 0 and d ≥ C log ℓ, we have After exponentiating and rearranging terms, we obtain M ℓ 2 < e bd(1−ϕ) , hence M ℓ 2 e −bd < e −bϕd .□ 4.4.Controlling pieces among many conjugacy classes.We can now apply Proposition 4.14 to bound the probability of a large J-loose cone-piece between a pair of conjugacy classes that are sampled from a large collection, where "large" is defined as a fraction of the systole.See Proposition 4.17.We can then combine this result with Proposition 4.7 to prove the main theorem.
Lemma 4.15.Fix constants q ∈ 0, 1 6 and J ∈ N. Let k ≤ e cℓ , where c < qb and b is the growth exponent of X. Select conjugacy classes [g 1 ], . . ., [g k ] at random from among those of length at most ℓ, with each [g i ] represented by a closed-geodesic w i .
Then, with overwhelming probability as ℓ → ∞, all J-loose cone-pieces among the w i have diameter strictly less than 2qℓ.
Proof.Since c < qb, we may choose a constant ϕ ∈ (0, 1) such that c < qbϕ 2 .We will consider pieces of diameter at least d ≥ 2qℓ.Since d is bounded below by a linear function of ℓ, the logarithmic hypothesis on d in Proposition 4.14 is satisfied for ℓ ≫ 0.
The above choices imply that there are k 2 ≤ e 2cℓ < e 2qbϕ 2 ℓ pairs of indices (i, j).When ℓ is sufficiently large, Proposition 4.14 says that for every pair (i, j), the probability of a J-loose cone-piece of diameter at least d between w i and w j is less than e −bdϕ .(This includes the case w i = w j .)Thus the total probability that some pair has a J-loose cone-piece of diameter ≥ d is less than k 2 e −bdϕ < e 2qbϕ 2 ℓ e −bdϕ = e (2qϕℓ−d) bϕ .
Recall that d ≥ 2qℓ and ϕ ∈ (0, 1).Then, as ℓ → ∞, the exponent in the above probability estimate is bounded as follows: We conclude that with overwhelming probability as ℓ → ∞, there are no J-loose pieces of diameter d ≥ 2qℓ.□ Lemma 4.16.Select conjugacy classes [g 1 ], . . ., [g k ] at random from among those of length at most ℓ, where k ≤ e cℓ for some constant c < qb where 0 < q < 1.Then, with overwhelming probability as ℓ → ∞, we have: Proof.Let w i be a closed-geodesic representing [g i ].By Theorem 2.4.(2), the conditional probability that Thus the conditional probability that |w i | < (1 − q)ℓ for some 1 ≤ i ≤ k is bounded above by (k Bℓ A )e −bqℓ .This upper bound approaches zero exponentially quickly when k ≤ e cℓ and c < qb.□ Proposition 4.17.Fix J ∈ N and α ∈ 0, 2 5 .Suppose conjugacy classes [g 1 ], . . ., [g k ] are chosen at random from among all those of length at most ℓ.Assume that k ≤ e cℓ for some constant c < bα/(α + 2).Then, with overwhelming probability as ℓ → ∞, all J-loose cone-pieces among the closed-geodesics representing [g 1 ], . . ., [g k ] have diameter strictly less than α|g i | for every i.
Proof.We will use Lemmas 4.15 and 4.16 with q = α α+2 .Observe that the hypothesis α ≤ 2 5 implies q ≤ 1 6 , as required for Lemma 4.15.With this value of q, Lemma 4.15 says that with overwhelming probability as ℓ → ∞, all cone-pieces have diameter strictly less than 2α α+2 ℓ.Meanwhile, Lemma 4.16 says that with overwhelming probability as ℓ → ∞, min k i=1 {|g i |} ≥ (1 − q)ℓ = 2 α+2 ℓ.Putting together the last two results, we conclude that the diameter of every cone-piece in w i is less than α|w i |. □ In the same spirit as Lemma 4.16, we have Lemma 4.18.Suppose k ≤ e cℓ for some constant c < b/2.Then, with overwhelming probability as ℓ → ∞, a set of k randomly chosen conjugacy classes has the property that each one is primitive.
Proof.By Remark 2.6, the probability that [g i ] is non-primitive is bounded above by Thus the probability that a k-tuple of conjugacy classes contains a non-primitive class is at most which goes to 0 as ℓ → ∞ because (c − b/2) < 0. □ We can now restate and prove Theorem 1.1.
Theorem 1.1.Let G = π 1 X, where X is a compact non-positively curved cube complex, and suppose that G is hyperbolic.Let b be the growth exponent of G with respect to X, and let a be the maximal growth exponent of a stabilizer of an essential hyperplane of X.Let k ≤ e cℓ , where Then with overwhelming probability as ℓ → ∞, for any set of conjugacy classes [g 1 ], . . ., [g k ] with each |g i | ≤ ℓ, the group G = G/⟨⟨g 1 , . . ., g k ⟩⟩ is hyperbolic and is the fundamental group of a compact, non-positively curved cube complex.
Proof.We may assume that G is non-elementary; otherwise, a = b = 0 and the theorem holds vacuously.We begin by replacing X with a closely related cube complex that satisfies Convention 4.3.First, we perform a cubical subdivision of X, while retaining the original metric.Then every conjugacy class [g] ⊂ G can be assigned a closed-geodesic representative w → X.Second, if some hyperplane of X is inessential, we replace X by its essential core, as in [CS11, Proposition 3.5], which has the same growth exponents a and b.For every relator [g i ] in the statement of the theorem, let w i → X be the chosen closedgeodesic.Let w i ⊂ X be a geodesic axis that covers w i , stabilized by g i ∈ [g i ].For each w i , let Y i = hull( w i ) ⊂ X, whose quotient Y i = ⟨g i ⟩\ Y i admits a local isometry into X.By Lemma 4.2, Y i lies in a uniform neighborhood of w i , hence its quotient Y i is compact and a quasi-circle.By construction, every hyperplane of Y i = hull( w i ) cuts w i .This gives us a cubical presentation Define J = J( X) as in Lemma 4.5.Then every diameter ≥ d cone-piece of X * between Y i and Y j corresponds to a J-loose cone-piece between w i and w j , of diameter ≥ d.Similarly, every diameter ≥ d wall-piece of X * in w i corresponds to a J-loose wall-piece, also of diameter ≥ d.
Next, we check the hypotheses of Theorem 3.5.We have verified that with overwhelming probability, X * is C ′ ( 1 20 ) and that every Y i is a compact quasi-circle that deformation retracts to a closed-geodesic w i .By Lemma 4.5.(3),every hyperplane U ⊂ Y i has a carrier N (U ) of diameter strictly less than α||Y i ||, which implies that N (U ) is embedded.Since U must cut the closed-geodesic w i , we also conclude that Y i ∖ U is contractible.Finally, Lemma 4.18 implies that with overwhelming probability, every w i is primitive.Therefore, Theorem 3.5 ensures that π 1 X * acts freely and cocompactly on the CAT(0) cube complex dual to the wallspace on X * .□

Generalization to other metric spaces
In this section, we prove Theorem 1.3, which generalizes Theorem 1.1 to the setting of groups acting on other, non-cubical metric spaces.The idea is to prove a probabilistic statement for quotients G = G/⟨⟨g 1 , . . ., g k ⟩⟩ where the relators g i are sampled from all short conjugacy classes in a G-action on some metric space Υ.As discussed in the introduction, there are many situations (for instance, hyperbolic manifolds) where the most natural geometry associated to a group G is carried by some metric space other than a cube complex.The results of this section enable us to draw conclusions using the growth of Υ rather than the growth of a cube complex.
Convention 5.1.The following assumptions and terminology hold throughout this section.Let G be a nonelementary, torsion-free group acting properly and cocompactly on a δhyperbolic geodesic metric space Υ.Let H 1 , . . ., H m be a collection of infinite index quasiconvex subgroups of G, which will remain fixed for the rest of this section.(In the main case of interest, the H i are hyperplane stabilizers of some action of G on a CAT(0) cube complex X.) Fix a basepoint υ ∈ Υ.
We assume that every nontrivial element g ∈ G stabilizes a geodesic axis γ ⊂ Υ.For every conjugacy class [g], we choose a representative closed-geodesic γ ⊂ G\Υ.Then every g ∈ [g] stabilizes some preimage γ of γ.Definition 5.2 (Loose pieces in Υ).Fix a constant J > 0. Consider bi-infinite geodesics γ, γ ′ ⊂ Υ that do not share an endpoint in ∂Υ.Observe that γ ∩ N J ( γ ′ ) is a closed set because N J ( γ ′ ) is a closed neighborhood, and is bounded because the geodesics do not share an endpoint.Thus γ ∩ N J ( γ ′ ) is compact.A J-loose cone-piece between γ and γ ′ is the maximal geodesic segment s ⊂ γ whose endpoints lie in N J ( γ ′ ).The companion of s is the maximal segment s ′ ⊂ γ ′ whose endpoints are at distance J from the corresponding endpoints of s.Now, let γ, γ ′ be closed-geodesics in G\Υ.Then a J-loose cone-piece between γ and γ ′ is the projection of a J-loose cone-piece between arbitrary preimages γ and γ ′ , excluding the case where γ = γ ′ .
A J-loose wall-piece in γ is a maximal geodesic segment s ⊂ γ whose endpoints are contained in N J (gH i p) for one of the chosen subgroups H i and for some g ∈ G.A J-loose wall-piece in a closed-geodesic γ is the projection of a J-loose wall-piece in γ.
Let g 1 , . . ., g k be infinite-order elements of G.For each g i , let γ i → G\Υ be the chosen closed-geodesic representing [g i ].Since each g i stabilizes an axis γ i ⊂ Υ, we have (1) For every g i , g j , the diameter of any 2δ-loose cone-piece between γ i and γ j is less than α g i Υ .
(2) For every g i , any κ-loose wall-piece in γ i has diameter less than α g i Υ .
The results of Section 4 have the following generalization to this context.Suppose that G acts properly and cocompactly on a δ-hyperbolic geodesic metric space Υ that admits a G-equivariant λ-quasiisometry from X.
Proof.We begin by recalling the above definition of a λ-quasiisometry.By Equation (1.1), there is a (G-equivariant) function f : X → Υ, along with positive constants λ 1 , λ 2 , ϵ such that λ 1 λ 2 = λ and We can now relate the systole of Y i in X to the (stable) translation length of g i in Υ.By Definition 3.2 and Definition 2.1, we have for an arbitrary point q ∈ X.
There is a constant η = η(δ, λ 1 , λ 2 , ϵ) with the following properties.First, for every convex set S ⊂ X, the image f (S) ⊂ Υ is η-quasiconvex.Second, for every bi-infinite geodesic w → X, the quasigeodesic image f ( w) is an η-fellow-traveler with every geodesic that has the same endpoints in ∂Υ.
Suppose that there is a diameter d cone-piece in X * between Y i and Y j , where d is very large (the precise criterion will be described below).Following Definition 3.2, this piece is a component of Y i ∩ Y j for appropriate preimages Y i and Y j .Let x, y ∈ Y i ∩ Y j be points such that d X (x, y) = d.By Lemma 4.2, there are points x i , y i ∈ w i and x j , y j ∈ w j that are K-close to x and y, respectively.
The following construction in Υ is illustrated in Figure 5.Let γ i , γ j be the representative axes for g i , g j , respectively, in Υ.Let ϖ i = f ( w i ) be the image of w i in Υ, and observe that ϖ i lies in the η-neighborhood of the axis γ i .Similarly, let ϖ j = f ( w j ), and observe that ϖ j lies in the η-neighborhood of the axis γ j .Label points , and similarly define x ′′ j , y ′′ j ∈ γ j .Observe that the second inequality in Equation (5.1) gives At the same time, the first inequality in Equation (5.1) gives . The figure lives in Υ.The blue quasi-geodesics ϖ i and ϖ j are the images of axes in X.The black geodesics are axes γ i and γ j , respectively.Since ϖ i must η-fellow-travel with γ i , there is a point x ′′ i that is η-close to x ′ i , and similarly for the others.The 2δ-loose cone-piece between γ i and γ j will contain the segment Thus, when d is sufficiently large, we have d Υ (x ′′ i , y ′′ i ) > 2λ 2 K + ϵ + 2η + 2δ.Next, observe that the geodesic quadrilateral with vertices x ′′ j , x ′′ i , y ′′ i , y ′′ j is 2δ-thin.Thus every point of [x ′′ i , y ′′ i ] ⊂ γ i must be 2δ-close to some other side of the quadrilateral.Let Since we have taken d to be large, this point x ′′′ i exists and is far from y ′′ i .Then the side that is 2δ-close to x ′′′ i must be on γ j .The same conclusion holds for the point Hence there is a 2δ-loose cone-piece in Υ, of diameter at least Combining the last computation with Equation (5.2), we obtain Since λ 1 λ 2 = λ < λ, there is a constant M 1 such that when g i Υ ≥ M 1 , we have In other words, the diameter of every cone-piece in Y i is less than α||Y i ||.This establishes the cone-piece portion of the claim that For the wall-piece portion of the desired conclusion, suppose that P = Y i ∩ N ( U j ), where U j is a hyperplane of X, and suppose that diam(P ) = d is very large.(Note that P may or may not be disjoint from Y i , hence might not be a wall-piece, according to Definition 3.2.Compare Remark 4.6.)Let x, y ∈ P be points realizing the diameter.By Lemma 4.2, there are points x i , y i ∈ w i that are K-close to x and y, respectively.Similarly, there are points x j , y j ∈ U j that are 1-close to x and y, respectively.
Let ϖ i = f ( w i ) be the image of w i in Υ, and observe that ϖ i lies in the η-neighborhood of the axis γ i .Let x ′ i = f (x i ), y ′ i = f (y i ) be points in ϖ i , and let x ′′ i , y ′′ i ∈ γ i be points within η of x ′ i , y ′ i .Let Θ j ⊂ Υ be the image of U j under the quasi-isometry, and let x ′ j = f (x j ), y ′ j = f (y j ) ∈ Θ j be the images of x j , y j .Recall that υ ∈ Υ is the pre-chosen basepoint, and that each orbit H j υ is κ j -quasiconvex.Note that Θ j ∈ N ψ (gH j υ) for some g ∈ G, where ψ depends on X, λ 1 , λ 2 , ϵ, κ j .Let x ′′ j , y ′′ j ∈ gH j υ be points that are ψ-close to x ′ j , y ′ j , respectively.First, observe that The same estimate holds for d Υ (y ′′ i , y ′′ j ).The geodesic quadrilateral with vertices x ′′ j , x ′′ i , y ′′ i , y ′′ j is 2δ-thin.Thus every point of [x ′′ i , y ′′ i ] ⊂ γ i must be 2δ-close to some other side of the quadrilateral.Let (As with cone pieces, such a point x ′′′ i exists whenever d is sufficiently large.)Then the side that is 2δ-close to x ′′′ i must be the opposite side, namely [x ′′ j , y ′′ j ].The same conclusion holds for the point The last containment uses the property κ ≥ 2δ + κ j , where gH j υ is κ j -quasiconvex.
Theorem 1.3.Let G = π 1 X, where X is a compact non-positively curved cube complex, and suppose that G is hyperbolic.Suppose that G also acts properly and cocompactly on a geodesic metric space Υ, where every non-trivial element of G stabilizes a geodesic axis.Suppose that there is a G-equivariant λ-quasiisometry X → Υ.
Let b be the growth exponent of G with respect to Υ, and let a be the maximal growth exponent in Υ of a stabilizer of an essential hyperplane of X.Let k ≤ e cℓ , where As in the proof of Theorem 1.1, we replace X by its essential core, so that all hyperplanes are essential.This does not affect the multiplicative constant λ in the quasi-isometry to Υ. Let H 1 , . . ., H m be the hyperplane stabilizers in G = π 1 X.We also subdivide X while retaining the original metric, so that every non-trivial conjugacy class is represented by a closedgeodesic.Both operations preserve the growth exponents a and b.Corollary 5.6 implies that with overwhelming probability as ℓ → ∞, the presentation ⟨G : H 1 , . . ., H m | g 1 , . . ., g k ⟩ is C ′ (α/λ) with respect to Υ and any choice of basepoint and axes.
For each i, let w i ⊂ X be a geodesic axis stabilized by For each w i , let Y i = hull( w i ) ⊂ X, whose quotient Y i = ⟨g i ⟩\ Y i admits a local isometry into X.By Lemma 4.2, Y i lies in a uniform neighborhood of w i , hence its quotient Y i is compact and a quasi-circle.By construction, every hyperplane of Y i cuts w i .This gives us a cubical presentation Now, Proposition 5.7 implies that with overwhelming probability as ℓ → ∞, the cubical presentation 20 .Furthermore, every hyperplane U of every Y i satisfies diam N (U ) < α||Y i ||.Since α < 1 14 , Lemma 3.4 ensures that π 1 X * is hyperbolic.
Next, we check the hypotheses of Theorem 3.5.We have verified that with overwhelming probability, X * is C ′ ( 1 20 ) and that every Y i is compact and deformation retracts to a closedgeodesic.By Proposition 5.7, every hyperplane U ⊂ Y i has a carrier N (U ) of diameter strictly less than α||Y i ||, which implies that N (U ) is embedded.Since U must cut the closedgeodesic w i , we also conclude that Y i ∖ U is contractible.Finally, the same argument as in Lemma 4.18 implies that with overwhelming probability, every g i is primitive.Therefore, Theorem 3.5 ensures that π 1 X * acts freely and cocompactly on the CAT(0) cube complex dual to the wallspace on X * .□ Remark 5.8.In the above proof of Theorem 1.3, all of the probabilistic arguments happen inside Corollary 5.6, which combines Propositions 5.4 and 5.5.By contrast, Proposition 5.7 involves a global assumption (a G-equivariant λ-quasiisometry), and the proof works entirely in the language of coarse geometry without invoking any counting or probability.One can envision a strengthening of Proposition 5.7 that tracks the behavior of a typical conjugacy class and a typical piece.
To make this precise, choose a basepoint x ∈ X and let B n ( X) = {g ∈ G : d X (x, gx) ≤ n}.Define B n (Υ) similarly, and recall from Definition 2.2 that f G,Υ (n) counts the cardinality of B n (Υ).Define the mean distortion of X with respect to Υ to be In words, τ ( X/Υ) measures the average factor by which an element of G sampled using Υ gets stretched in X. See [CT21, Eqn (1.1) and Thm 1.2] for a proof that the limit exists.
The mean distortion can be bounded as follows.Suppose there is a G-equivariant λ-quasiisometry f : X → Υ, as in Equation (5.1), so that f (x) = υ.Then For a chosen ϵ > 0, an element g ∈ G or a conjugacy class [g] is called Υ-typical if For each ϵ > 0, a conjugacy class sampled uniformly from all those of Υ-length at most ℓ will be Υ-typical with overwhelming probability.We now discuss the prospects for removing λ from the statement of Proposition 5.7.The proof begins by observing that every conjugacy class [g i ] satisfies g i X ≥ (λ 2 ) −1 g i Υ ; see Equation (5.2).Since [g i ] is Υ-typical with overwhelming probability, we may replace (λ 2 ) −1 by τ ( X/Υ) − ϵ .Next, the proof of Proposition 5.7 uses the constant λ 1 to pass from the diameter of a piece in X * to the diameter of a loose piece in Υ; see Equation (5.3).If one knew that a cone-piece or wall-piece in X * , coming from a random presentation as in the statement of Theorem 1.3, also corresponds to an Υ-typical group element of G, one could replace λ 1 by τ ( X/Υ) + ϵ .The upshot would be that the product λ = λ 1 λ 2 would be replaced by the quotient τ ( X/Υ) + ϵ / τ ( X/Υ) − ϵ , which approaches 1 as ϵ → 0.
Whether the pieces coming from a random presentation (sampled using lengths in Υ) can be represented by Υ-typical group elements is an interesting problem.See Question 7.10.

Pentagonal surfaces
Throughout this section, we consider the setting where Υ = H 2 is the hyperbolic plane, equipped with a tiling T by regular right-angled pentagons, and X is the square complex dual to this tiling.We work out the optimal constant λ in a λ-quasiisometry from X to Υ.This can be considered the first interesting example where Theorem 1.3 applies.Proposition 6.1.Let T be the tiling of H 2 by regular right-angled pentagons, with hyperbolic metric d H .The dual tiling T * , with five quadrilaterals at every vertex, can be identified with a CAT(0) square complex X, with combinatorial metric d X .Then the identity map id : Furthermore, this value of λ is optimal.
As mentioned in the introduction, combining Theorem 1.3 with Proposition 6.1 and the classical work of Huber [Hub59] yields a proof of Corollary 1.4.
The proof of Proposition 6.1 is entirely elementary and largely pictorial; see Figures 6-13.The proof also trades the coarse geometry that has dominated most of this paper for the fine geometry of H 2 .We begin with Lemma 6.2, which computes a number of lengths in a single right-angled pentagon using the hyperbolic laws of sines and cosines [Fen89,Rat19].We then continue with Proposition 6.3, which considers a number of combinatorial possibilities for how a hyperbolic geodesic segment can cross a sequence of several adjacent pentagons.Lemma 6.2.Let P ⊂ H 2 be a regular right-angled pentagon.Label lengths as in Figure 6.Set K = cos 2π 5 .Then the labeled lengths can be expressed as follows: Furthermore, the length g of a geodesic segment contained in two adjacent right-angled pentagons, as in Figure 6, satisfies 2g = arccosh(1 + 2K(8K 2 + 8K + 1) 2 ) ≈ 3.1838.(6.7) Proof.In Figure 6, left, the right-angled pentagon P is subdivided into five isometric quadrilaterals, arranged symmetrically about the center point of P .Each quadrilateral Q has two sides of length a, two sides of length b, three right angles, and one angle of θ = hence K + 1 = cosh 2 a, implying Equation (6.1).By [Rat19, Thm 3.5.8],we have implying Equation (6.2).The hyperbolic law of cosines [Rat19, Thm 3.5.3]implies the following version of the Pythagorean theorem as a special case.In a hyperbolic right triangle, with legs of length x, y and hypotenuse of length z, the lengths satisfy (6.8) cosh x • cosh y = cosh z.
Now, consider the distances between the midpoints of edges in the right-angled pentagon P .By (6.8), the midpoints of two adjacent edges are separated by distance c, where cosh c = cosh 2 a = sinh 2 a + 1 = K + 1, implying Equation (6.3).By [Rat19, Thm 3.5.3], the midpoints of two non-adjacent edges are separated by distance d, where implying Equation (6.4).
Finally, we use Figure 6 to compute the length g.In that figure, we have a hyperbolic quadrilateral with two right angles at the ends of a side of length 2a, adjacent sides of length 4a, and the fourth side of length 2g.According to first formula in the last block of displayed equations on [Fen89, page 88], we have (6.9)cosh(2g) = − sinh 2 (4a) + cosh 2 (4a) cosh(2a).
Substituting all this into Equation (6.9) gives cosh implying Equation (6.7).□ For the rest of this section, the symbols a, . . ., g will always denote the constants computed in the above lemma.Now, we can prove the following more global comparison between the cubical and hyperbolic metrics on H 2 .Proposition 6.3.Let T be the tiling of H 2 by regular right-angled pentagons, with hyperbolic metric d H .The dual tiling T * , with five quadrilaterals at every vertex, can be identified with a CAT(0) square complex X, with combinatorial metric d X .Then there is a constant ϵ > 0 such that for all x, y ∈ X 0 , the distances d H (x, y) and d X (x, y) can be compared as follows: where the constants c, d are as in Lemma 6.2.Furthermore, the multiplicative constants c, d in (6.10) are sharp.
Observe that the constant λ in the statement of Proposition 6.1 is exactly d/c.Thus Proposition 6.3 implies Proposition 6.1.
Proof.We begin by proving the second inequality of (6.10).Let w → X be a combinatorial geodesic in X with endpoints x, y ∈ X 0 .By choosing a sufficiently large additive constant ϵ, we may assume without loss of generality that |w| = d X (x, y) ≥ 2. In the hyperbolic metric on H 2 , the combinatorial geodesic w is a concatenation of two or more edges of the dual tiling T * .By Lemma 6.2, every edge of w has hyperbolic length 2b.
Since edges of T * meet at angles of θ = 2π 5 or 2θ = 4π 5 , we may homotope w to a shorter piecewise-geodesic path γ by constructing hyperbolic shortcuts between midpoints of consecutive edges.See  By the same token, sharpness of the multiplicative constant c in the first inequality of (6.10) holds because one may concatenate arbitrarily many segments of length c in adjacent pentagons to form a hyperbolic geodesic, as in the right panel of Figure 7.
. The two combinatorial types of segment s cutting through a right-angled pentagon, and the altitude ℓ altitude associated to each s.
To prove the first inequality of (6.10), with its optimal multiplicative constant, we make the following definitions.An altitude of a right-angled pentagon P is a geodesic segment ℓ from a vertex to the midpoint of the opposite edge.If s ⊂ P is a hyperbolic geodesic connecting interior points of sides E 1 , E 2 , the altitude associated to s is the unique altitude ℓ with the property that reflection in ℓ interchanges E 1 with E 2 .See Figure 8.
Let γ → H 2 be a hyperbolic geodesic with endpoints x, y ∈ X 0 .Then d X (x, y) equals the number of hyperplanes that separate x from y. Since the hyperplanes of X are identified with the bi-infinite geodesics containing edges of T , it follows that d X (x, y) is the number n of edges of T crossed by γ.After an arbitrarily small perturbation, affecting len H (γ) by an additive error, we may assume that γ is disjoint from T 0 .
In every pentagon P that intersects γ but does not contain the endpoints x, y, draw the altitude associated to γ ∩ P .These (n − 1) altitudes partition γ into a concatenation γ 1 γ 2 • • • γ n , where γ 1 is the segment from x to the first altitude; γ n is the segment from the last altitude to y; and every remaining γ i connects the altitudes in adjacent right-angled pentagons.To prove (6.10), we will show that the average length of a segment γ i for 0 < i < n is at least c ≈ 0.7672. .The five combinatorial types of (unoriented) intersection between a hyperbolic geodesic and two adjacent pentagons P, P ′ .In types I-IV, the geodesic γ enters through a side of P that is not adjacent to the shared side P ∩ P ′ .In type V, the geodesic γ both enters and exits P ∪ P ′ through sides adjacent to the shared side P ∩ P ′ .
There are five combinatorial possibilities for a segment γ i between two altitudes, corresponding to five types of intersections between γ and two adjacent pentagons.See Figure 9.If ℓ and ℓ ′ are the altitudes in adjacent pentagons P and P ′ , respectively, the convexity of distance functions implies that the shortest geodesic segment from ℓ to ℓ ′ meets each of them orthogonally or at an endpoint.Consequently, it is not hard to determine the shortest possible length from ℓ to ℓ ′ ; see Figure 10.In type I, Figure 10 shows that the worst-case scenario for len(γ i ) is a ≈ 0.5306; since a < c, this complicated case is analyzed below.In type II, Figure 10 shows that we always have len(γ i ) ≥ 2a > c.In type III, we always have len(γ i ) ≥ d > c.In type IV, we always have len(γ i ) ≥ f /2 > c.Finally, in type V, we always have len(γ i ) ≥ c.The comparisons to c come from the numerical values computed in Lemma 6.2.Thus, in every case except type I, we have len(γ i ) ≥ c.
It remains to analyze segments γ i of type I.We claim the following: (1) If γ i is of type I, and 2 < i < n − 1, then there is an adjacent index j = i ± 1 such that the corresponding segment γ j is of type II, III, or IV.(2) If γ j is of type II or IV, then γ i is the only type-I segment adjacent to γ j .
(3) For all types of γ j , we have len(γ i ) + len(γ j ) ≥ min(3a, g) > 2c.(4) If γ j is of type III, and furthermore γ j also adjacent to a type-I segment γ k where k = i ± 2, then len(γ i ) + len(γ j ) + len(γ k ) ≥ 2e > 3c.Assuming these claims, we can complete the proof as follows.By Claim 1, every segment γ i of type I (where 3 ≤ i ≤ n − 2) borrows some of the length from an adjacent segment γ j of type II, III, or IV.By Claim 2, every segment γ j of type II or IV acts as a "lender" to at most one segment of type I. Every segment γ j of type III acts as a "lender" to at most two segments of type I.In all cases, Claims 3 and 4 say that the average length of γ j and its adjacent type-I segments is more than c.Putting it all together, we have Figure 11 illustrates the proof of Claims 1 and 2. The geodesic γ intersects the left-most pair of pentagons {P, Q} in a segment γ i ⊂ γ of type I.We assume without loss of generality that the indices are increasing as γ traverses the figure from left to right.The continuation of γ must exit the central pentagon R through one of the sides marked II, III, or IV, because γ cannot intersect δ twice.This proves Claim 1.If the next segment γ j = γ i+1 is of type II, then the following segment γ k = γ i+2 cannot be of type I, because otherwise γ would again intersect δ twice.Similarly, if γ j = γ i+1 is of type IV, then the following segment γ k = γ i+2 cannot be of type I, because γ cannot intersect the geodesic ξ twice.This proves Claim 2.
For the three combinatorial types of γ j = γ i+1 , Figure 12 illustrates the infimal lengths of (γ i ∪ γ j ).Let ℓ II , ℓ III , ℓ IV be the terminal altitudes for the three possible types of γ j .If γ j is of type II, then the worst-case scenario is when γ fellow-travels δ, hence we obtain II III IV . The left-most segment γ i is of type I.The next segment γ j = γ i+1 is of type II, III, or IV, and terminates at the altitude ℓ II , ℓ III , or ℓ IV , respectively.In type II and type IV, the shortest possible lengths of γ i ∪ γ i+1 are highlighted.In type III, γ i+1 must intersect ℓ II or ℓ IV .len(γ i ∪ γ j ) > 3a > 2c, where the final inequality uses Lemma 6.2.If γ j is of type IV, then the worst-case scenario is when γ i ∪ γ j starts at the endpoint of an altitude and ends perpendicular to ℓ IV .In this case, we have len(γ i ∪ γ j ) ≥ g > 2c, where the final inequality is by Lemma 6.2.Finally, observe that if γ j is of type III, then γ j must intersect either ℓ II or ℓ IV .Thus, by the cases already discussed, we have len(γ i ∪ γ j ) > 2c.This proves Claim 3.
Figure 13.If γ j = γ i+1 is of type III and γ i , γ i+2 are both of type I, the shortest possible configuration is symmetric about the point in the center.In this configuration, each half of (γ i ∪ γ i+1 ∪ γ i+2 ) is a segment of length e.
Finally, Figure 13 illustrates the proof of Claim 4. If γ j = γ i+1 is of type III and γ i , γ i+2 are both of type I, then the shortest possible length of (γ i ∪ γ j ∪ γ k ) is 2e > 3c, where the inequality follows from Lemma 6.2.This completes the proof of the Claims, implying Equation (6.10) and the Proposition.□

Problems
This section collects several problems and suggested directions for future research.
Problem 7.1.Determine the optimal density at which random quotients of a surface group G are cubulated.How does the answer depend on the choice of proper metric on G?
Problem 7.2.Given a non-positively curved cube complex X such that G = π 1 X is hyperbolic, find the optimal density such that quotients of G are cubulated.How does the answer depend on X and the growth rate of its hyperplanes?
The following three problems are stated in order of increasing relatively hyperbolic ambition.
Problem 7.3.Let G = G 1 * G 2 be a free product of cubulated groups.Find the optimal density at which random quotients of G are cubulated.Letting G i = π 1 X i , where X i is a non-positively curved cube complex, we think of G = π 1 X, where X = X 1 ∪ A ∪ X 2 , where A ∼ = [0, n] is an arc glued to X 1 and X 2 at its endpoints.Does the optimal density depend on the length of A? See Martin-Steenbock [MS17] and Jankiewicz-Wise [JW22].
Problem 7.4.Let G be a relatively hyperbolic group that is the fundamental group of a compact non-positively curved cube complex.Show that there is a density at which random quotients of G are cubulated (and relatively hyperbolic).
Problem 7.5.Let G be an acylindrically hyperbolic group that is the fundamental group of a compact non-positively curved cube complex.Show that there is a density at which random quotients of G are cubulated.
The following problem has not even been studied for virtually free groups: Problem 7.6.Generalize Theorem 1.1 to cubulated hyperbolic groups with torsion.
The main challenge is that cubical small-cancellation theory was not described in the case where there is torsion.One could either develop that theory, or use another hyperbolic Dehn filling theory with walls (e.g.Osin [Osi07] or Groves-Manning [GM08] or Dahmani-Guirardel-Osin [DGO17]) to ensure that the relators embed in the universal cover of the quotient, and subsequently apply the criterion of Theorem 3.5 which can instead be applied to X * .)Problem 7.7.Let G be a non-elementary relatively hyperbolic group.Prove that there is a density such that all low-density quotients of G are again relatively hyperbolic.Note that the (torsion-free) hyperbolic case was already handled by Ollivier [Oll04].
An unlikely but more fanciful possibility is to use Theorem 1.1 to find new examples of hyperbolic groups that are not cubulated.
Conjecture 7.8.Let M = H n /Γ be a closed hyperbolic n-manifold.Assume that either n ≤ 3 or that M is arithmetic of simplest type.Then, for every λ > 1, M is homotopy equivalent to a compact non-positively curved cube complex X such that there is a Γ-equivariant λ-quasiisometry from X to M .
Recall that the definition of a λ-quasiisometry appears in Equation (1.1).Conjecture 7.8 is supported by the following intuition.If M is a surface, then there is a plethora of cubulations of M via closed-geodesics.As ℓ → ∞, a randomly chosen closed-geodesic γ of length approximately ℓ is nearly equidistributed in M (as well as in T 1 M ).Therefore, the number of lifts of γ separating a distant pair of points p, q ∈ M = H 2 should be nearly proportional to d H 2 (p, q).The same intuition applies for n = 3, where a closed hyperbolic 3-manifold M has a plethora of cubulations via nearly geodesic surfaces that are similarly nearly equidistributed in M .See Kahn and Markovic [KM12].

Figure 1 .
Figure1.A cubical presentation illustrating conditions (1) and (2) of Definition 3.6.In this example, as in Theorem 3.5, we have π 1 Y i ∼ = Z for each Y i , and the wallspace structure on Y i has two diametrically opposed hyperplanes in each wall.Unlike the setting of Theorem 3.5, π 1 X is not hyperbolic in this example.
Remark 3.7 ([Wis21, Rem 5.43]).Let M be the maximal diameter of any piece of Y in X * .Then a hyperplane u of Y is piecefully convex provided its carrier N = N (u) satisfies: d Y (g N , N ) > M for any translate g N ̸ = N ⊂ Y .The following is a simplified restatement of [Wis21, Thm 5.44], which also incorporates [Wis21, Lem 3.70 and Cor 5.45].
Since x, hx ∈ N r ( v), considering closest-point projections to v gives d(x, hx) ≤ r + h + r ≤ 2r + L. Meanwhile Equation (4.3) gives d(hx, y) ≤ L. Thus R = 2r + 2L is a uniform bound on |u| = d(x, y).Since L ≥ 7J by definition, we have R ≥ 14J.□ Now, we argue that closed-geodesics containing a piece of diameter ≥ d happen with a probability that decays exponentially with d.

Figure 7 .
Each such shortcut replaces two cubical half-edges (of combined cubical length 1) by a hyperbolic segment of length either c or d.The first and last half-edges of w remain as they are, and have hyperbolic length b.Since c < d, it follows that d H 2 (x, y) ≤ b + d(|w| − 1) + b ≤ d|w| + ϵ = d • d X (x, y) + ϵ, for an appropriate value of ϵ.Sharpness of the multiplicative constant d holds because one may concatenate arbitrarily many segments of length d in adjacent pentagons to form a hyperbolic geodesic.See the left panel of Figure 7.

Figure 7 .
Figure 7.The blue path w → X is a combinatorial geodesic from x to y.The pink path γ is constructed by taking hyperbolic shortcuts between midpoints of consecutive edges of w.Each segment of γ in a pentagon of T has length c or d.The sharpness of the constants c and d in (6.10) is demonstrated by the right and left panels, respectively.
Figure9.The five combinatorial types of (unoriented) intersection between a hyperbolic geodesic and two adjacent pentagons P, P ′ .In types I-IV, the geodesic γ enters through a side of P that is not adjacent to the shared side P ∩ P ′ .In type V, the geodesic γ both enters and exits P ∪ P ′ through sides adjacent to the shared side P ∩ P ′ .

Figure 10 .
Figure10.For each combinatorial type of intersection between a hyperbolic geodesic and two adjacent pentagons, the highlighted segment shows the infimal length between the altitudes.

d H 2 Figure 11 .
Figure 11.Three possibilities for a geodesic γ that intersects {P, Q} in a segment of type I.The next segment of γ in pentagons Q ∪ R must be of type II, III, or IV.
Indeed, if Υ is the Cayley graph of G with respect to some generating set, this follows from a large deviation result of Cantrell and Tanaka [CT21, Thm 4.23].If Υ is itself a CAT(0) cube complex, this follows from a large deviation theorem of Cantrell and Reyes [CR23b, Thm 1.4].For general Υ, the referee informs us that this can be derived from Cantrell and Reyes [CR23a, Eqn (5.2)].