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# Coxeter polytopes and Benjamini-Schramm convergence

Jean Raimbault

► To cite this version:

Jean Raimbault. Coxeter polytopes and Benjamini-Schramm convergence. Bulletin de la société mathématique de France, 2024, 151 (1), pp.33-44. 10.24033/bsmf.2884. hal-04625566

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https://hal.science/hal-04625566

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# Coxeter polytopes and Benjamini-Schramm convergence

Jean Raimbault

► To cite this version:

Jean Raimbault. Coxeter polytopes and Benjamini-Schramm convergence. Bulletin de la société mathématique de France, 2024, 151 (1), pp.33-44. 10.24033/bsmf.2884. hal-04625566

HAL Id: hal-04625566

https://hal.science/hal-04625566v1

Submitted on 26 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
COXETER POLYTOPES AND BENJAMINI-SCHRAMM
CONVERGENCE

JEAN RAIMBAULT

ABSTRACT. We observe that a large part of the volume of a hyperbolic polyhedron is taken by a tubular neighbourhood of its boundary, and use this to give a new proof for the finiteness of arithmetic maximal reflection groups following a recent work with M. Frączyk and S. Hurtado. We also study in more depth the case of polygons in the hyperbolic plane.

Let $X$ be a space of constant curvature, that is either a hyperbolic space $\mathbb{H}^d$, a Euclidean space $\mathbb{R}^d$ or a sphere $\mathbb{S}^d$. An hyperplane in $X$ is a one-lower-dimensional complete totally geodesic subspace, and a polytope is a bounded (or in the case of $\mathbb{H}^d$, finite-volume) region delimited by a finite number of hyperplanes. A polytope in $X$ is said to be Coxeter if the dihedral angles between its faces are each of the form $\pi/m$ for some $m \ge 2$. For Coxeter polytopes in $\mathbb{R}^d$ or $\mathbb{S}^d$ there is a well-known, complete and very intelligible classification of Coxeter polytopes by Coxeter diagrams. On the other hand Coxeter polytopes in $\mathbb{H}^d$ have a very different behaviour and are still quite mysterious. In the sequel we will thus be only concerned with $X = \mathbb{H}^d$. We will study Coxeter polytopes from a metric viewpoint and establish some results about their shapes when the volume tends to infinity, especially when $d=2$. A general survey on Coxeter groups in hyperbolic space is given in [Vin85]; a more recent one is [Bel16] which focuses on arithmetic aspects.

Let $P$ be a Coxeter polytope in $\mathbb{H}^d$, and let $\Gamma_P$ be the subgroup of $\mathrm{Isom}(\mathbb{H}^d)$ generated by reflections in the faces of $P$. This is a discrete subgroup acting on $X$ with fundamental domain $P$, by the Poincaré polyhedron theorem. Moreover, $P$ is identified with the $\mathbb{H}^d$-orbifold given by $\Gamma_P \backslash \mathbb{H}^d$, by endowing each face of $P$ with the local orbifold structure given by its pointwise stabiliser (the group generated by reflexions in the maximal faces which contain it). For $R > 0$ the $R$-thin part of $P$ is given by

$$P_{\le R} = \{x \in P : \exists \gamma \in \Gamma_P, d(x, \gamma x) \le R/2\}$$

and it corresponds to the $R$-thin part of the orbifold. An easy exercise shows that $P_{\le R}$ is equal to the set of points in $P$ which are at distance at most $R/2$ from the boundary. The first result in this note is the following, which is essentially a consequence of the hyperbolic isoperimetric inequality as we prove in 1 below.
COXETER POLYTOPES AND BENJAMINI-SCHRAMM
CONVERGENCE

JEAN RAIMBAULT

ABSTRACT. We observe that a large part of the volume of a hyperbolic polyhedron is taken by a tubular neighbourhood of its boundary, and use this to give a new proof for the finiteness of arithmetic maximal reflection groups following a recent work with M. Frączyk and S. Hurtado. We also study in more depth the case of polygons in the hyperbolic plane.

Let $X$ be a space of constant curvature, that is either a hyperbolic space $\mathbb{H}^d$, a Euclidean space $\mathbb{R}^d$ or a sphere $\mathbb{S}^d$. An hyperplane in $X$ is a one-lower-dimensional complete totally geodesic subspace, and a polytope is a bounded (or in the case of $\mathbb{H}^d$, finite-volume) region delimited by a finite number of hyperplanes. A polytope in $X$ is said to be Coxeter if the dihedral angles between its faces are each of the form $\pi/m$ for some $m \ge 2$. For Coxeter polytopes in $\mathbb{R}^d$ or $\mathbb{S}^d$ there is a well-known, complete and very intelligible classification of Coxeter polytopes by Coxeter diagrams. On the other hand Coxeter polytopes in $\mathbb{H}^d$ have a very different behaviour and are still quite mysterious. In the sequel we will thus be only concerned with $X = \mathbb{H}^d$. We will study Coxeter polytopes from a metric viewpoint and establish some results about their shapes when the volume tends to infinity, especially when $d = 2$. A general survey on Coxeter groups in hyperbolic space is given in [Vin85]; a more recent one is [Bel16] which focuses on arithmetic aspects.

Let $P$ be a Coxeter polytope in $\mathbb{H}^d$, and let $\Gamma_P$ be the subgroup of $\mathrm{Isom}(\mathbb{H}^d)$ generated by reflections in the faces of $P$. This is a discrete subgroup acting on $X$ with fundamental domain $P$, by the Poincaré polyhedron theorem. Moreover, $P$ is identified with the $\mathbb{H}^d$-orbifold given by $\Gamma_P \backslash \mathbb{H}^d$, by endowing each face of $P$ with the local orbifold structure given by its pointwise stabiliser (the group generated by reflexions in the maximal faces which contain it). For $R > 0$ the $R$-thin part of $P$ is given by

$$P_{\le R} = \{x \in P : \exists \gamma \in \Gamma_P, d(x, \gamma x) \le R/2\}$$

and it corresponds to the $R$-thin part of the orbifold. An easy exercise shows that $P_{\le R}$ is equal to the set of points in $P$ which are at distance at most $R/2$ from the boundary. The first result in this note is the following, which is essentially a consequence of the hyperbolic isoperimetric inequality as we prove in 1 below.
**Theorem 1.** For every $d \ge 2$ there exists a constant $C(d)$ such that for every Coxeter polytope $P$ of finite volume in $\mathbb{H}^d$ we have

$$ \mathrm{vol}(P_{\le 2}) \ge C(d) \mathrm{vol}(P). $$

In a joint work with M. Frączyk and S. Hurtado [FHR23] it was proven that $\mathrm{vol}(M_{\le R}) = o(\mathrm{vol} M)$ uniformly for $M$ a congruence arithmetic orbifold quotient of a given symmetric space. From this together with Theorem 1 we can quite easily deduce the following result, which was originally proved by Nikulin and Agol-Belolipetsky-Storm-Whyte ([Nik07, ABSW08] respectively). In fact the inspiration for this note was provided by a recent work of Fisher-Hurtado [FH22] where they use a lower-level part of [FHR23]<fnref n="1" /> to give a new proof of Nikulin and Agol-Belolipetsky-Storm-Whyte’s result.

**Corollary 2.** For any $d$ there are at most finitely many arithmetic maximal reflection groups in $\mathrm{PO}(d, 1)$.

*Proof.* Let $\Gamma_{P_n}$ be a sequence of pairwise nonconjugated maximal Coxeter arithmetic lattices, that is $P_n$ is a finite-volume Coxeter polyhedron in $\mathbb{H}^d$ and $\mathrm{vol}(P_n) \to +\infty$. Let $\Gamma_n$ be the congruence closure of $\Gamma_{P_n}$. Theorem D in [FHR23] states that if $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \to +\infty$ then, putting $M_n = \Gamma_n \backslash \mathbb{H}^d$, we have that for any $R > 0$, $\lim_{n \to +\infty} \frac{\mathrm{vol}(M_n)_{< R}}{\mathrm{vol} M_n} = 0$. Since $P_n$ is a finite (orbifold) cover of $M_n$ and the ratio $\frac{\mathrm{vol}(\cdot)_{< R}}{\mathrm{vol} \cdot}$ is decreasing in finite covers, this would contradict the fact that $\mathrm{vol}(P_n)_{\le 2} \ge C(d) \mathrm{vol}(P_n)$ if we knew that $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \to +\infty$.

So, to deduce the corollary we need only prove that $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \to +\infty$. This is the case if the adjoint trace fields of the $\Gamma_{P_n}$ are not of bounded degree. On the other hand, if these trace fields have bounded degree then by [ABSW08, Lemma 6.2] and standard arguments (see e.g. Lemma 5.4 in loc. cit.) we have that $[\Gamma_n : \Gamma_{P_n}]$ is bounded, so that $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \gg \mathrm{vol}(P_n)$ goes to infinity. □

It is also well-known that cofinite reflection groups cannot exist in large dimensions [Vin81, Kho86], so we may as well say that there are only finitely many congruence (or maximal arithmetic) hyperbolic reflection groups.

In the case of polygons in $\mathbb{H}^2$ we can say more. Let $G = \mathrm{Isom}(\mathbb{H}^d) = \mathrm{PO}(d, 1)$, let $\mu_P$ be the $G$-invariant measure on the Chabauty space $\mathrm{Sub}_G$ of $G$ supported on the conjugacy class of $\Gamma_P$. We will use the notion of Benjamini-Schramm convergence introduced in [ABB$^+$17, Sections 2-3]; this is the notion of convergence induced by the topology of weak convergence of measures on $\mathrm{Sub}_G$. In this language Theorem 1 implies that the trivial subgroup is not a limit point of the measures $\mu_P$. When $d = 2$ we can prove the following result, which is much more precise than Theorem 1.

<footnote n="1">¹Namely the “arithmetic Margulis lemma”, Theorem 3.1 in loc. cit. which is an essential ingredient in the proof of Theorem D in loc. cit.</footnote>
**Theorem 1.** For every $d \ge 2$ there exists a constant $C(d)$ such that for every Coxeter polytope $P$ of finite volume in $\mathbb{H}^d$ we have

$$ \mathrm{vol}(P_{\le 2}) \ge C(d) \mathrm{vol}(P). $$

In a joint work with M. Frączyk and S. Hurtado [FHR23] it was proven that $\mathrm{vol}(M_{\le R}) = o(\mathrm{vol} M)$ uniformly for $M$ a congruence arithmetic orbifold quotient of a given symmetric space. From this together with Theorem 1 we can quite easily deduce the following result, which was originally proved by Nikulin and Agol-Belolipetsky-Storm-Whyte ([Nik07, ABSW08] respectively). In fact the inspiration for this note was provided by a recent work of Fisher-Hurtado [FH22] where they use a lower-level part of [FHR23]<fnref n="1" /> to give a new proof of Nikulin and Agol-Belolipetsky-Storm-Whyte's result.

**Corollary 2.** For any $d$ there are at most finitely many arithmetic maximal reflection groups in $\mathrm{PO}(d, 1)$.

*Proof.* Let $\Gamma_{P_n}$ be a sequence of pairwise nonconjugated maximal Coxeter arithmetic lattices, that is $P_n$ is a finite-volume Coxeter polyhedron in $\mathbb{H}^d$ and $\mathrm{vol}(P_n) \to +\infty$. Let $\Gamma_n$ be the congruence closure of $\Gamma_{P_n}$. Theorem D in [FHR23] states that if $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \to +\infty$ then, putting $M_n = \Gamma_n \backslash \mathbb{H}^d$, we have that for any $R > 0$, $\lim_{n \to +\infty} \frac{\mathrm{vol}(M_n)_{< R}}{\mathrm{vol} M_n} = 0$. Since $P_n$ is a finite (orbifold) cover of $M_n$ and the ratio $\frac{\mathrm{vol}(\cdot)_{< R}}{\mathrm{vol}}$ is decreasing in finite covers, this would contradict the fact that $\mathrm{vol}(P_n)_{\le 2} \ge C(d) \mathrm{vol}(P_n)$ if we knew that $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \to +\infty$.

So, to deduce the corollary we need only prove that $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \to +\infty$. This is the case if the adjoint trace fields of the $\Gamma_{P_n}$ are not of bounded degree. On the other hand, if these trace fields have bounded degree then by [ABSW08, Lemma 6.2] and standard arguments (see e.g. Lemma 5.4 in loc. cit.) we have that $[\Gamma_n : \Gamma_{P_n}]$ is bounded, so that $\mathrm{vol}(\Gamma_n \backslash \mathbb{H}^d) \gg \mathrm{vol}(P_n)$ goes to infinity. □

It is also well-known that cofinite reflection groups cannot exist in large dimensions [Vin81, Kho86], so we may as well say that there are only finitely many congruence (or maximal arithmetic) hyperbolic reflection groups.

In the case of polygons in $\mathbb{H}^2$ we can say more. Let $G = \mathrm{Isom}(\mathbb{H}^d) = \mathrm{PO}(d, 1)$, let $\mu_P$ be the $G$-invariant measure on the Chabauty space $\mathrm{Sub}_G$ of $G$ supported on the conjugacy class of $\Gamma_P$. We will use the notion of Benjamini-Schramm convergence introduced in [ABB$^+$17, Sections 2-3]; this is the notion of convergence induced by the topology of weak convergence of measures on $\mathrm{Sub}_G$. In this language Theorem 1 implies that the trivial subgroup is not a limit point of the measures $\mu_P$. When $d = 2$ we can prove the following result, which is much more precise than Theorem 1.

<footnote n="1">¹Namely the “arithmetic Margulis lemma”, Theorem 3.1 in loc. cit. which is an essential ingredient in the proof of Theorem D in loc. cit.</footnote>
**Theorem 3.** If $\mu$ belongs to the closure (in topology of weak convergence of measures on $\text{Sub}_G$) of the set of all $\mu_P$ for $P$ a finite-volume Coxeter polygon in $\mathbb{H}^2$ then $\mu$-almost every subgroup is non-trivial and generated by reflections.

This seems likely to be true in higher dimensions as well, though our very elementary proof does not seem to immediately extend to this setting. Finally, note that any subgroup of $\text{Isom}(\mathbb{H}^d)$ generated by reflections is in fact generated by the reflections in the side of a Coxeter polytope (possibly with infinitely many faces), a well-known fact<fnref n="2" />

**Organisation.** The proof of Theorem 1 is very short and given in Section 1. The rest of the article is dedicated to the proof of Theorem 3; first we collect a few useful facts on the geometry of hyperbolic Coxeter polygons in Section 2, and use them in Section 3 to prove that a Benjamini–Schramm limit of Coxeter polygons is almost surely non-trivial. Independently, we prove in Section 4 that the set of groups generated by reflections is closed in the Chabauty topology and deduce that a Benjamini–Schramm limit of Coxeter polygons is almost surely generated by reflections.

**Acknowledgments.** Research supported by ANR grant AGDE - ANR-20-CE40-0010-01. I am grateful to David Fisher for comments on earlier versions of the paper, and to a referee for pointing out several imprecisions in the submitted version.

## 1. PROOF OF THEOREM 1

Fix $d \ge 2$. Let $P$ be a Coxeter polytope in $\mathbb{H}^d$. We will denote by $F_i, i \in I$ the $(d-1)$-faces of $P$.

Now let

$$U = \{x \in P : d(x, \partial P) \le 1\}$$

and

$$P' = \{x \in P : d(x, \partial P) \ge 1\}.$$

We have $U \subset P_{\le 2}$ since every point of $U$ is moved by at most 2 by a reflection in a face of $P$ (as previously noted, it is clear that in fact $U = P_{\le 2}$).

For $x \in F_i$ let $\nu_x$ be the vector normal to $F_i$ pointing inside $P$; note that $d(F_i, \exp_x(\nu_x)) = 1$ for all $x \in F_i$. Let $W_i \subset F_i$ defined by

$$W_i = \{x \in F_i : \exp_x(\nu_x) \in P', \forall j \neq i \text{ d}(F_j, \exp_x(\nu_x)) > 1\}$$

and let

$$F'_i = \{\exp_x(\nu_x) : x \in W_i\}$$

Then the $F'_i$ are disjoint open subsets of $\partial P'$ and their complement $S = \partial P' \cup \bigcup_{i \in I} F'_i$ is of measure 0 (with respect to the $(d-1)$-dimensional measure on $\partial P'$) as it is equal to the set of points in $\partial P'$ at distance 1 from at least two of the $F_i$.

<footnote n="2">²This is stated at the beginning of [Vin85], and the proof is more or less obvious.</footnote>
**Theorem 3.** If $\mu$ belongs to the closure (in topology of weak convergence of measures on $\text{Sub}_G$) of the set of all $\mu_P$ for $P$ a finite-volume Coxeter polygon in $\mathbb{H}^2$ then $\mu$-almost every subgroup is non-trivial and generated by reflections.

This seems likely to be true in higher dimensions as well, though our very elementary proof does not seem to immediately extend to this setting. Finally, note that any subgroup of $\text{Isom}(\mathbb{H}^d)$ generated by reflections is in fact generated by the reflections in the side of a Coxeter polytope (possibly with infinitely many faces), a well-known fact<fnref n="2" />

**Organisation.** The proof of Theorem 1 is very short and given in Section 1. The rest of the article is dedicated to the proof of Theorem 3; first we collect a few useful facts on the geometry of hyperbolic Coxeter polygons in Section 2, and use them in Section 3 to prove that a Benjamini-Schramm limit of Coxeter polygons is almost surely non-trivial. Independently, we prove in Section 4 that the set of groups generated by reflections is closed in the Chabauty topology and deduce that a Benjamini-Schramm limit of Coxeter polygons is almost surely generated by reflections.

**Acknowledgments.** Research supported by ANR grant AGDE - ANR-20-CE40-0010-01. I am grateful to David Fisher for comments on earlier versions of the paper, and to a referee for pointing out several imprecisions in the submitted version.

## 1. PROOF OF THEOREM 1

Fix $d \ge 2$. Let $P$ be a Coxeter polytope in $\mathbb{H}^d$. We will denote by $F_i, i \in I$ the $(d-1)$-faces of $P$.

Now let

$$U = \{x \in P : d(x, \partial P) \le 1\}$$

and

$$P' = \{x \in P : d(x, \partial P) \ge 1\}.$$

We have $U \subset P_{\le 2}$ since every point of $U$ is moved by at most 2 by a reflection in a face of $P$ (as previously noted, it is clear that in fact $U = P_{\le 2}$).

For $x \in F_i$ let $\nu_x$ be the vector normal to $F_i$ pointing inside $P$; note that $d(F_i, \exp_x(\nu_x)) = 1$ for all $x \in F_i$. Let $W_i \subset F_i$ defined by

$$W_i = \{x \in F_i : \exp_x(\nu_x) \in P', \forall j \neq i \text{ d}(F_j, \exp_x(\nu_x)) > 1\}$$

and let

$$F'_i = \{\exp_x(\nu_x) : x \in W_i\}$$

Then the $F'_i$ are disjoint open subsets of $\partial P'$ and their complement $S = \partial P' \cup \bigcup_{i \in I} F'_i$ is of measure 0 (with respect to the $(d-1)$-dimensional measure on $\partial P'$) as it is equal to the set of points in $\partial P'$ at distance 1 from at least two of the $F_i$.

<footnote n="2">²This is stated at the beginning of [Vin85], and the proof is more or less obvious.</footnote>
For $y \in F'_i$ let $\nu'_y$ be the vector orthogonal to $F'_i$ pointing outside $P'$; then the map

$$ (1.1) \qquad E: [0,1] \times \partial P' \setminus S \to \mathbb{H}^d, (t, y) \mapsto \exp_y(t\nu'_y) $$

has its image inside $U$. Since the local geometry of the submanifolds $F'_i$ of $\mathbb{H}^d$ depends only on $d$, we see that the Jacobian of $E$ is uniformly bounded away from 0 (we prove this in detail in 1.1 at the end of the section); let $\varepsilon(d) > 0$ be a lower bound. Moreover, the sets $E([0, 1] \times F'_i)$ are pairwise disjoint (since a point in $E([0, 1] \times F'_i)$ is at distance $\le 1$ from exactly 1 face of $\partial P$, which is $F_i$, and at distance $> 1$ of all others). It follows that

$$ \mathrm{vol}_d(E([0,1] \times \partial P' \setminus S)) \ge \varepsilon(d) \cdot 1 \cdot \mathrm{vol}_{d-1}(\partial P' \setminus S) = \varepsilon(d) \mathrm{vol}_{d-1}(\partial P') $$

so that $\mathrm{vol}(U) \ge \varepsilon(d) \mathrm{vol}_{d-1}(\partial P')$. We finish the proof of the theorem with the following chain of inequalities:

$$ \begin{aligned} \mathrm{vol}_d(P) &= \mathrm{vol}_d(U) + \mathrm{vol}_d(P') \\ &\le \mathrm{vol}_d(U) + \mathrm{vol}_{d-1}(\partial P') \\ &\le (1 + \varepsilon(d)^{-1}) \mathrm{vol}_d(U) \\ &\le C(d) \mathrm{vol}_d(P_{\le 2}). \end{aligned} $$

where the second inequality follows from the isoperimetric inequality for hyperbolic space [Gro07, Proposition 6.6] which implies that $\mathrm{vol}_d(P') \le \mathrm{vol}_{d-1}(\partial P')$.

1.1. **Exponential map on equidistant sets.** Let $H$ be a geodesic hyperplane in $\mathbb{H}^d$ and $H'$ a connected component of $\{x \in \mathbb{H}^d : d(x, H) = 1\}$. Let $E: [0, 1] \times H' \to \mathbb{H}^d$ be the map defined as in (1.1). Since all hyperplanes and their equidistant sets are related by isometries, and exponential maps are equivariant with respect to those, our claim will follow if prove that the Jacobian $\det(DE(x, t))$ is uniformly bounded away from 0 for $x \in H'$, $t \in [0, 1]$.

The group $\mathrm{Isom}(H)$ acts transitively on $H'$ and the map $E$ is equivariant with respect to this action. It follows that we need only to prove that $\det(DE(t, x))$ is uniformly bounded away from 0 for a fixed $x$ and $t \in [0, 1]$. This is immediate by compacity, since $DE(t, x)$ is invertible for all $t \in \mathbb{R}$.

## 2. LEMMAS ON COXETER POLYGONS

We collect here some preliminary facts about Coxeter polygons in $\mathbb{H}^2$, and give complete proofs for all of them; though they are likely well-known it seems more convenient to give their (short) proofs than locate sufficiently precise references for them. First we have a consequence of the collar/Margulis lemma.

**Lemma 4.** There exists $\eta > 0$ such that if $P$ is a Coxeter polygon in $\mathbb{H}^2$ then:

(1) if an edge of $P$ has length $\le \eta$ then its adjacent angles are right angles;
For $y \in F'_i$ let $\nu'_y$ be the vector orthogonal to $F'_i$ pointing outside $P'$; then the map

$$ (1.1) \qquad E: [0,1] \times \partial P' \setminus S \to \mathbb{H}^d, (t, y) \mapsto \exp_y(t\nu'_y) $$

has its image inside $U$. Since the local geometry of the submanifolds $F'_i$ of $\mathbb{H}^d$ depends only on $d$, we see that the Jacobian of $E$ is uniformly bounded away from 0 (we prove this in detail in 1.1 at the end of the section); let $\varepsilon(d) > 0$ be a lower bound. Moreover, the sets $E([0, 1] \times F'_i)$ are pairwise disjoint (since a point in $E([0, 1] \times F'_i)$ is at distance $\le 1$ from exactly 1 face of $\partial P$, which is $F_i$, and at distance $> 1$ of all others). It follows that

$$ \mathrm{vol}_d(E([0,1] \times \partial P' \setminus S)) \ge \varepsilon(d) \cdot 1 \cdot \mathrm{vol}_{d-1}(\partial P' \setminus S) = \varepsilon(d) \mathrm{vol}_{d-1}(\partial P') $$

so that $\mathrm{vol}(U) \ge \varepsilon(d) \mathrm{vol}_{d-1}(\partial P')$. We finish the proof of the theorem with the following chain of inequalities:

$$ \begin{aligned} \mathrm{vol}_d(P) &= \mathrm{vol}_d(U) + \mathrm{vol}_d(P') \\ &\le \mathrm{vol}_d(U) + \mathrm{vol}_{d-1}(\partial P') \\ &\le (1 + \varepsilon(d)^{-1}) \mathrm{vol}_d(U) \\ &\le C(d) \mathrm{vol}_d(P_{\le 2}). \end{aligned} $$

where the second inequality follows from the isoperimetric inequality for hyperbolic space [Gro07, Proposition 6.6] which implies that $\mathrm{vol}_d(P') \le \mathrm{vol}_{d-1}(\partial P')$.

1.1. **Exponential map on equidistant sets.** Let $H$ be a geodesic hyperplane in $\mathbb{H}^d$ and $H'$ a connected component of $\{x \in \mathbb{H}^d : d(x, H) = 1\}$. Let $E: [0, 1] \times H' \to \mathbb{H}^d$ be the map defined as in (1.1). Since all hyperplanes and their equidistant sets are related by isometries, and exponential maps are equivariant with respect to those, our claim will follow if prove that the Jacobian $\det(DE(x, t))$ is uniformly bounded away from 0 for $x \in H'$, $t \in [0, 1]$.

The group $\mathrm{Isom}(H)$ acts transitively on $H'$ and the map $E$ is equivariant with respect to this action. It follows that we need only to prove that $\det(DE(t, x))$ is uniformly bounded away from 0 for a fixed $x$ and $t \in [0, 1]$. This is immediate by compacity, since $DE(t, x)$ is invertible for all $t \in \mathbb{R}$.

## 2. LEMMAS ON COXETER POLYGONS

We collect here some preliminary facts about Coxeter polygons in $\mathbb{H}^2$, and give complete proofs for all of them; though they are likely well-known it seems more convenient to give their (short) proofs than locate sufficiently precise references for them. First we have a consequence of the collar/Margulis lemma.

**Lemma 4.** There exists $\eta > 0$ such that if $P$ is a Coxeter polygon in $\mathbb{H}^2$ then:

(1) if an edge of $P$ has length $\le \eta$ then its adjacent angles are right angles;
(2) no two adjacent edges of P have both length $\le \eta$;

(3) any two non-consecutive vertices of P are at distance at least $\eta$.

*Proof.* Let $\Gamma = \Gamma_P$ be the discrete subgroup generated by the reflections $\sigma_e$ in the sides $e$ of P. Let $\delta$ be the constant given by the collar/Margulis lemma for $\mathbb{H}^2$, so that for any $x \in \mathbb{H}^2$ we have that

$$ \Gamma_x := \langle \gamma \in \Gamma : d(x, \gamma x) \le \delta \rangle $$

is virtually cyclic. So if $e_2$ is an edge of P with length $\le \delta/2$ and $e_1, e_3$ the adjacent edges of P then the subgroup generated by $\sigma_{e_i}$ is virtually cyclic (as each of $\sigma_{e_i}$ moves any vertex of $e$ of less than $\delta$). On the other hand discrete virtually cyclic subgroups of $PGL_2(\mathbb{R})$ cannot contain an element of finite order other than 2 (such a subgroup contains an hyperbolic isometry and any finite-order element in the group must preserve the 2 endpoints of its axis), and as this subgroup contains the rotations about the vertices of $e_2$, the angles between $e_1, e_1$ and $e_2, e_3$ must be right angles. This proves part 1 for any $\eta \le \delta/2$.

We now prove part 2 for any $\eta \le \delta/4$. Assume that there are two consecutive edges with lengths less than $\eta$, i.e. three consecutive vertices $x_1, x_2, x_3$ such that $d(x_1, x_2), d(x_2, x_3) \le \eta$. Then if $\sigma_i, 1 \le i \le 4$ are the reflections in the sides containing the $x_j$, each $x_j$ is at distance less than $2\eta$ of the axis of each $\sigma_i$ so $d(x_j, \sigma_i x_j) < 4\eta \le \delta$. By the Collar Lemma it follows that the subgroup generated by the reflections $\sigma_i$ must be virtually cyclic. On the other hand it contains three rotations whose centers (namely $x_1, x_2, x_3$) are not colinear, which is impossible if it is virtually cyclic.

To prove the last point let $v, w$ be two non-consecutive edges of P and let $e'$ be the common perpendicular between $v$ and $w$. Since P is a Coxeter polygon the sum of the angles of P at $v$ and $w$ is at most $\pi$, so there are two edges $e_1, e_2$ of P on the same side of $e'$ such that the sum of the angles between $e'$ and $e_1, e_2$ is at most $\pi/2$. Let $\beta, \gamma$ be these angles. Since the hyperbolic triangle with angles $\pi/4, \beta, \gamma$ has area at least $\pi/4$, at least one of its edges is of length $\ge d$ where $d$ is the smallest diameter of a hyperbolic disc of area $\ge \pi/4$. Either it is the side length $a$ opposite to the angle $\pi/4$, or we can assume that it is the side length $b$ opposite to the angle $\beta$ and in this case we have $\sinh(a) = \frac{2\sqrt{2}\sinh(b)}{\sin(\beta)} \ge 2\sqrt{2}\sinh(r)$. In any case we get that the side adjacent to the angles $\beta, \gamma$ in this triangle has length bounded below by a constant $a_0$ independent of $\beta, \gamma$. It follows that if $v, w$ are at distance less than $a_0$ then the half-lines supported by $e_1, e_2$ must intersect in $\mathbb{H}^2$. It follows that the subgroup generated by the reflections in the edges of P adjacent to $v, w$ contain at least 3 rotations with non-aligned center, so by Margulis lemma we must have $d(v, w) \ge \delta/2$. This proves that 3 holds for any $\eta \le \min(\delta/2, a_0)$. $\square$

Next we will need the following lemma on the area of hyperbolic triangles.
(2) no two adjacent edges of P have both length $\le \eta$;

(3) any two non-consecutive vertices of P are at distance at least $\eta$.

*Proof.* Let $\Gamma = \Gamma_P$ be the discrete subgroup generated by the reflections $\sigma_e$ in the sides $e$ of P. Let $\delta$ be the constant given by the collar/Margulis lemma for $\mathbb{H}^2$, so that for any $x \in \mathbb{H}^2$ we have that

$$ \Gamma_x := \langle \gamma \in \Gamma : d(x, \gamma x) \le \delta \rangle $$

is virtually cyclic. So if $e_2$ is an edge of P with length $\le \delta/2$ and $e_1, e_3$ the adjacent edges of P then the subgroup generated by $\sigma_{e_i}$ is virtually cyclic (as each of $\sigma_{e_i}$ moves any vertex of $e$ of less than $\delta$). On the other hand discrete virtually cyclic subgroups of $PGL_2(\mathbb{R})$ cannot contain an element of finite order other than 2 (such a subgroup contains an hyperbolic isometry and any finite-order element in the group must preserve the 2 endpoints of its axis), and as this subgroup contains the rotations about the vertices of $e_2$, the angles between $e_1, e_1$ and $e_2, e_3$ must be right angles. This proves part 1 for any $\eta \le \delta/2$.

We now prove part 2 for any $\eta \le \delta/4$. Assume that there are two consecutive edges with lengths less than $\eta$, i.e. three consecutive vertices $x_1, x_2, x_3$ such that $d(x_1, x_2), d(x_2, x_3) \le \eta$. Then if $\sigma_i, 1 \le i \le 4$ are the reflections in the sides containing the $x_j$, each $x_j$ is at distance less than $2\eta$ of the axis of each $\sigma_i$ so $d(x_j, \sigma_i x_j) < 4\eta \le \delta$. By the Collar Lemma it follows that the subgroup generated by the reflections $\sigma_i$ must be virtually cyclic. On the other hand it contains three rotations whose centers (namely $x_1, x_2, x_3$) are not colinear, which is impossible if it is virtually cyclic.

To prove the last point let $v, w$ be two non-consecutive edges of P and let $e'$ be the common perpendicular between $v$ and $w$. Since P is a Coxeter polygon the sum of the angles of P at $v$ and $w$ is at most $\pi$, so there are two edges $e_1, e_2$ of P on the same side of $e'$ such that the sum of the angles between $e'$ and $e_1, e_2$ is at most $\pi/2$. Let $\beta, \gamma$ be these angles. Since the hyperbolic triangle with angles $\pi/4, \beta, \gamma$ has area at least $\pi/4$, at least one of its edges is of length $\ge d$ where $d$ is the smallest diameter of a hyperbolic disc of area $\ge \pi/4$. Either it is the side length $a$ opposite to the angle $\pi/4$, or we can assume that it is the side length $b$ opposite to the angle $\beta$ and in this case we have $\sinh(a) = \frac{2\sqrt{2}\sinh(b)}{\sin(\beta)} \ge 2\sqrt{2}\sinh(r)$. In any case we get that the side adjacent to the angles $\beta, \gamma$ in this triangle has length bounded below by a constant $a_0$ independent of $\beta, \gamma$. It follows that if $v, w$ are at distance less than $a_0$ then the half-lines supported by $e_1, e_2$ must intersect in $\mathbb{H}^2$. It follows that the subgroup generated by the reflections in the edges of P adjacent to $v, w$ contain at least 3 rotations with non-aligned center, so by Margulis lemma we must have $d(v, w) \ge \delta/2$. This proves that 3 holds for any $\eta \le \min(\delta/2, a_0)$. $\square$

Next we will need the following lemma on the area of hyperbolic triangles.
**Lemma 5.** For any $l_0$ there exists a constant $A > 0$ such that any hyperbolic triangle with edges of length at least $l_0$ and angles at most $\pi/2$ has area at least $A$.

*Proof.* Let $a, b, c$ be the edge lengths of such a triangle $T$ and define

$$s = \frac{a+b+c}{2}, \quad s_a = s-a, \quad s_b = s-b, \quad s_c = s-c.$$

The hyperbolic Heron formula [Med12, Theorem 1.1(i)] states that

$$\sin(\text{Area}(T)) = \frac{\sqrt{\sinh(s) \sinh(s_a) \sinh(s_b) \sinh(s_c)}}{4 \cosh(a/2) \cosh(b/2) \cosh(c/2)}.$$

Using the hyperbolic cosine law it follows from our hypotheses on $T$ that there exists a constant $l_1 > 0$ (independent of $a, b, c$) such that $s, s_a, s_b, s_c \ge l_1$. It follows that we have $\sinh(s_a) \gg e^{s_a}$ and similarly for the other terms. As $s_a + s_b + s_c = 2s$ we get that

$$\sinh(s) \sinh(s_a) \sinh(s_b) \sinh(s_c) \gg e^{3s}.$$

Similarly,

$$\cosh(a/2) \cosh(b/2) \cosh(c/2) \ll e^s$$

so in the end

$$\text{Area}(T) \ge \sin(\text{Area}(T)) \gg e^{s/2}$$

which finishes the proof. $\square$

<figure><img src="image_3.png" /></figure>

Finally we will use the following lemma.

**Lemma 6.** Let $\varepsilon > 0$. There exists a constant $\eta' > 0$ such that for any Coxeter polygon $P$ in $\mathbb{H}^2$ and any $R$ there exists a polygon $P'$ such that

(1) $P'$ has no edge length smaller than $\eta'$;

(2) $\frac{\text{Area}(P'_{\le R})}{\text{Area} P'} \le 2 \frac{\text{Area}(P_{\le R})}{\text{Area} P}$.

*Proof.* We construct $P'$ as follows: let $v_1, \dots, v_m$ be a cyclic ordering of the vertices of $P$. Let $\eta$ be the constant from Lemma 4, and $0 < \alpha < 1/2$ (to be determined later). For each pair $(v_i, v_{i+1})$ of adjacent vertices such that $d(v_i, v_{i+1}) \le \alpha \cdot \eta$ we remove the vertex $v_{i+1}$ from $P$, that is, if $i_1, \dots, i_k$ are those indices such that $d(v_{i_j}, v_{i_{j+1}}) \le \alpha \cdot \eta$ we take $P'$ to be the polygon spanned by the $v_i, i \notin \{i_1, \dots, i_k\}$. Since $d(v_{i_{j+1}}, v_{i_{j+2}}) > \eta$ by Lemma 4, it follows from the triangle inequality that $d(v_{i_j}, v_{i_{j+2}}) > (1-\alpha)\eta > \alpha\eta$ and so $P'$ satisfies condition 1 for any $\eta' \le \alpha\eta$.

We now prove that $P'$ satisfies condition 2 for sufficiently small $\alpha$. First we estimate the area of each removed triangle. To do this let $i$ such that $d(v_i, v_{i+1}) < \alpha\eta$; we want to estimate the area of the triangle $T_i$ spanned by
**Lemma 5.** For any $l_0$ there exists a constant $A > 0$ such that any hyperbolic triangle with edges of length at least $l_0$ and angles at most $\pi/2$ has area at least $A$.

*Proof.* Let $a, b, c$ be the edge lengths of such a triangle $T$ and define

$$s = \frac{a+b+c}{2}, \quad s_a = s-a, \quad s_b = s-b, \quad s_c = s-c.$$

The hyperbolic Heron formula [Med12, Theorem 1.1(i)] states that

$$\sin(\text{Area}(T)) = \frac{\sqrt{\sinh(s) \sinh(s_a) \sinh(s_b) \sinh(s_c)}}{4 \cosh(a/2) \cosh(b/2) \cosh(c/2)}.$$

Using the hyperbolic cosine law it follows from our hypotheses on $T$ that there exists a constant $l_1 > 0$ (independent of $a, b, c$) such that $s, s_a, s_b, s_c \ge l_1$. It follows that we have $\sinh(s_a) \gg e^{s_a}$ and similarly for the other terms. As $s_a + s_b + s_c = 2s$ we get that

$$\sinh(s) \sinh(s_a) \sinh(s_b) \sinh(s_c) \gg e^{3s}.$$

Similarly,

$$\cosh(a/2) \cosh(b/2) \cosh(c/2) \ll e^s$$

so in the end

$$\text{Area}(T) \ge \sin(\text{Area}(T)) \gg e^{s/2}$$

which finishes the proof. $\square$

<figure><img src="image_2.png" /></figure>

Finally we will use the following lemma.

**Lemma 6.** Let $\varepsilon > 0$. There exists a constant $\eta' > 0$ such that for any Coxeter polygon $P$ in $\mathbb{H}^2$ and any $R$ there exists a polygon $P'$ such that

(1) $P'$ has no edge length smaller than $\eta'$;

(2) $\frac{\text{Area}(P'_{\le R})}{\text{Area} P'} \le 2\frac{\text{Area}(P_{\le R})}{\text{Area} P}$.

*Proof.* We construct $P'$ as follows: let $v_1, \dots, v_m$ be a cyclic ordering of the vertices of $P$. Let $\eta$ be the constant from Lemma 4, and $0 < \alpha < 1/2$ (to be determined later). For each pair $(v_i, v_{i+1})$ of adjacent vertices such that $d(v_i, v_{i+1}) \le \alpha \cdot \eta$ we remove the vertex $v_{i+1}$ from $P$, that is, if $i_1, \dots, i_k$ are those indices such that $d(v_{i_j}, v_{i_{j+1}}) \le \alpha \cdot \eta$ we take $P'$ to be the polygon spanned by the $v_i, i \notin \{i_1, \dots, i_k\}$. Since $d(v_{i_{j+1}}, v_{i_{j+2}}) > \eta$ by Lemma 4, it follows from the triangle inequality that $d(v_{i_j}, v_{i_{j+2}}) > (1-\alpha)\eta > \alpha\eta$ and so $P'$ satisfies condition 1 for any $\eta' \le \alpha\eta$.

We now prove that $P'$ satisfies condition 2 for sufficiently small $\alpha$. First we estimate the area of each removed triangle. To do this let $i$ such that $d(v_i, v_{i+1}) < \alpha\eta$; we want to estimate the area of the triangle $T_i$ spanned by
$v_i, v_{i+1}, v_{i+2}$. Let $\gamma$ be its angle at $v_i$, $c = d(v_{i+1}, v_{i+2})$, $a = d(v_i, v_{i+2})$; by the hyperbolic sine law we have that $\sinh(a) = \frac{\sinh(c)}{\sin(\gamma)}$. We compute that

$$ \sin(\gamma) \ge \frac{\sinh(a - \eta)}{\sinh(a)} \\ \ge \frac{\sinh(a) - \eta \cosh(a)}{\sinh(a)} \ge 1 - u \cdot \alpha $$

where $u > 0$ on the last line depends only on $\eta$. It follows that $\gamma \ge \frac{\pi}{2} - u' \cdot \alpha$, and finally that

$$ (2.1) \qquad \text{Area}(T_i) \le u' \cdot \alpha $$

for some $u'$ independent of $P, \alpha$.

Now the triangle $S_i$ spanned by $v_{i-1}, v_i, v_{i+2}$ has all its edge length at least $\eta$, by Lemma 4. Its angles are at most $\pi/2$ since it is inscribed in the Coxeter polygon $P$. So by Lemma 5 we have that $\text{Area}(S_i) \ge A$. By (2.1) this implies that $\text{Area}(T_i) = O(\text{Area}(S_i)\alpha)$ uniformly in $i, P$, and as $\text{Area}(P') = \text{Area}(P) - \sum_{l=1}^k \text{Area}(T_{i_l})$ it follows that

$$ \text{Area}(P') = (1 - O(\alpha)) \text{Area}(P). $$

On the other hand, we have that $P'_{\le R}$ is contained in the $\alpha\eta$-neighbourhood of $P_{\le R}$ so that

$$ \text{Area}((P')_{\le R} \le (1 + O(\alpha)) \text{Area}(P_{\le R}). $$

From these two inequalities it follows that

$$ \frac{\text{Area}((P')_{\le R})}{\text{Area}(P')} \le (1 + O(\alpha)) \frac{\text{Area}(P_{\le R})}{\text{Area}(P)} $$

which finishes the proof by taking $\eta' = \alpha\eta$ for $\alpha$ small enough (independently of $P, R$).

□

### 3. NONTRIVIALITY OF BS-LIMITS OF POLYGONS

In this section we give the proof of the first part of Theorem 3, that a Benjamini-Schramm limit of a sequence of Coxeter polygons is almost surely nontrivial. We will prove that for any sequence $P_n$ of Coxeter polygons in $\mathbb{H}^2$ we have

$$ (3.1) \qquad \lim_{R \to +\infty} \lim_{n \to +\infty} \frac{\text{Area}((P_n)_{\ge R})}{\text{vol} P_n} = 0. $$

from which the first statement follows immediately.

Let $P'_n$ be the polygons obtained from the $P_n$ by applying Lemma 6; it follows from the condition (1) that it suffices to prove (3.1) for the $P'_n$. Note that in any triangulation<fnref n="3" /> of $P'_n$ all angles are at most $\pi/2$ (they are smaller

<footnote n="3">³By this we mean a decomposition of a polygon into triangles whose vertices are vertices of the polygon.</footnote>
$v_i, v_{i+1}, v_{i+2}$. Let $\gamma$ be its angle at $v_i$, $c = d(v_{i+1}, v_{i+2})$, $a = d(v_i, v_{i+2})$; by the hyperbolic sine law we have that $\sinh(a) = \frac{\sinh(c)}{\sin(\gamma)}$. We compute that

$$ \sin(\gamma) \ge \frac{\sinh(a - \eta)}{\sinh(a)} \\ \ge \frac{\sinh(a) - \eta \cosh(a)}{\sinh(a)} \ge 1 - u \cdot \alpha $$

where $u > 0$ on the last line depends only on $\eta$. It follows that $\gamma \ge \frac{\pi}{2} - u' \cdot \alpha$, and finally that

$$ (2.1) \qquad \text{Area}(T_i) \le u' \cdot \alpha $$

for some $u'$ independent of $P, \alpha$.

Now the triangle $S_i$ spanned by $v_{i-1}, v_i, v_{i+2}$ has all its edge length at least $\eta$, by Lemma 4. Its angles are at most $\pi/2$ since it is inscribed in the Coxeter polygon $P$. So by Lemma 5 we have that $\text{Area}(S_i) \ge A$. By (2.1) this implies that $\text{Area}(T_i) = O(\text{Area}(S_i)\alpha)$ uniformly in $i, P$, and as $\text{Area}(P') = \text{Area}(P) - \sum_{l=1}^k \text{Area}(T_{i_l})$ it follows that

$$ \text{Area}(P') = (1 - O(\alpha)) \text{Area}(P). $$

On the other hand, we have that $P'_{\le R}$ is contained in the $\alpha\eta$-neighbourhood of $P_{\le R}$ so that

$$ \text{Area}((P')_{\le R} \le (1 + O(\alpha)) \text{Area}(P_{\le R}). $$

From these two inequalities it follows that

$$ \frac{\text{Area}((P')_{\le R})}{\text{Area}(P')} \le (1 + O(\alpha)) \frac{\text{Area}(P_{\le R})}{\text{Area}(P)} $$

which finishes the proof by taking $\eta' = \alpha\eta$ for $\alpha$ small enough (independently of $P, R$).

□

### 3. NONTRIVIALITY OF BS-LIMITS OF POLYGONS

In this section we give the proof of the first part of Theorem 3, that a Benjamini-Schramm limit of a sequence of Coxeter polygons is almost surely nontrivial. We will prove that for any sequence $P_n$ of Coxeter polygons in $\mathbb{H}^2$ we have

$$ (3.1) \qquad \lim_{R \to +\infty} \lim_{n \to +\infty} \frac{\text{Area}((P_n)_{\ge R})}{\text{vol} P_n} = 0. $$

from which the first statement follows immediately.

Let $P'_n$ be the polygons obtained from the $P_n$ by applying Lemma 6; it follows from the condition (1) that it suffices to prove (3.1) for the $P'_n$. Note that in any triangulation<fnref n="3" /> of $P'_n$ all angles are at most $\pi/2$ (they are smaller

<footnote n="3">³By this we mean a decomposition of a polygon into triangles whose vertices are vertices of the polygon.</footnote>
<figure><img src="image_4.png" /><figcaption>FIGURE 1. Triangulations of ideal n-gons and their dual tree</figcaption></figure>

than those of the $P_n$ and the latter are of the form $\pi/k, k \ge 2$), and by using
point (3) of Lemma 4 in addition we see that all edges in the triangulation
have length at least $\eta$, so by Lemma 5 we have a uniform lower bound for
all areas of triangles occuring in any triangulation of any $P'_n$.

We triangulate $P'_n$ as follows : we choose a vertex, and as long as possible add an edge between the current vertex and the second-to-next one (clockwise). See Figure 1 for an illustration.

The tree $T_n$ dual to this triangulation, rooted at the last triangle, has radius at most $\log_2(n) + 1$ by [KLM$^+$18, Proposition 2.1]. Moreover the distance of the closest leaf to this triangle is at least $\log_2(n) - 1$ as well$^4$.

Let $S > 0$. Let $(T_n)_{\le S}$ be the set of vertices in $T_n$ which are at distance at most $S$ from $\partial T_n^5$. Then $T_n \setminus (T_n)_{\le S}$ is contained in the ball of radius $\log_2(n) + 1 - S$ since every leaf is at distance at most $\log_2(n) + 1$ from the root. It follows that $|T_n \setminus (T_n)_{\le S}| \le 2^{\log_2(n)+1-S} = O(2^{-S}|T_n|)$ since $|T_n| = n$. So we have

$$
(3.2) \qquad \left| (T_n)_{\le S} \right| \ge (1 - O(2^{-S})) |T_n|.
$$

On the other hand, if $x \in P'_n$ lies in a triangle corresponding to a triangle
$t \in (T_n)_{\le S}$ then we may construct a path $c$ as follows: let $t = t_0$; choose a
side of $t_0$ closest to $x$ such that the corresponding edge of $T_n$ points away
from the root, and let $c_0$ be the geodesic from $x$ to this side. If the side lies
on the boundary let $c = c_0$; otherwise let $t_1$ be the triangle on the other
side, let $x_1$ be the foot of $c_0$ on the side $t_0 \cap t_1$ and iterate the construction
until the boundary is reached, say in $l$ steps, and let $c$ be the concatenation
of $c_0, \dots, c_l$ where $c_i$ is the path obtained at the $(i+1)$th step. As $t \in$

<footnote>$^4$This can easily be seen by observing that these functions are monotonic, and for a 2$^k$-gon thay are equal to $\log_2(n) \pm 1$ (it would be more natural in this case to center at the edge between the last two triangles).</footnote>

<footnote n="5">⁵For us the boundary of a finite tree is its set of leaves.</footnote>
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Triangulations of ideal n-gons and their dual tree</figcaption></figure>

than those of the $P_n$ and the latter are of the form $\pi/k$, $k \ge 2$), and by using
point (3) of Lemma 4 in addition we see that all edges in the triangulation
have length at least $\eta$, so by Lemma 5 we have a uniform lower bound for
all areas of triangles occuring in any triangulation of any $P'_n$.

We triangulate $P'_n$ as follows : we choose a vertex, and as long as possible add an edge between the current vertex and the second-to-next one (clockwise). See Figure 1 for an illustration.

The tree $T_n$ dual to this triangulation, rooted at the last triangle, has radius at most $\log_2(n) + 1$ by [KLM$^+$18, Proposition 2.1]. Moreover the distance of the closest leaf to this triangle is at least $\log_2(n) - 1$ as well$^4$.

Let $S > 0$. Let $(T_n)_{\le S}$ be the set of vertices in $T_n$ which are at distance at most $S$ from $\partial T_n^5$. Then $T_n \setminus (T_n)_{\le S}$ is contained in the ball of radius $\log_2(n) + 1 - S$ since every leaf is at distance at most $\log_2(n) + 1$ from the root. It follows that $|T_n \setminus (T_n)_{\le S}| \le 2^{\log_2(n)+1-S} = O(2^{-S}|T_n|)$ since $|T_n| = n$. So we have

$$
(3.2) \qquad \left| (T_n)_{\le S} \right| \ge (1 - O(2^{-S})) |T_n|.
$$

On the other hand, if $x \in P'_n$ lies in a triangle corresponding to a triangle
$t \in (T_n)_{\le S}$ then we may construct a path $c$ as follows: let $t = t_0$; choose a
side of $t_0$ closest to $x$ such that the corresponding edge of $T_n$ points away
from the root, and let $c_0$ be the geodesic from $x$ to this side. If the side lies
on the boundary let $c = c_0$; otherwise let $t_1$ be the triangle on the other
side, let $x_1$ be the foot of $c_0$ on the side $t_0 \cap t_1$ and iterate the construction
until the boundary is reached, say in $l$ steps, and let $c$ be the concatenation
of $c_0, \dots, c_l$ where $c_i$ is the path obtained at the $(i+1)$th step. As $t \in$

<footnote>$^4$This can easily be seen by observing that these functions are monotonic, and for a 2$^k$-gon thay are equal to $\log_2(n) \pm 1$ (it would be more natural in this case to center at the edge between the last two triangles).</footnote>

<footnote n="5">⁵For us the boundary of a finite tree is its set of leaves.</footnote>
$(T_n)_{\le S}$ and the edges $(t_i, t_{i+1})$ point away from the root we must have that $l \le S + 1$. Moreover, the length of each $c_i$ is at most $\log(3)$ since $\mathbb{H}^2$ is $\log(3)/2$-hyperbolic (which implies that in any geodesic triangle any point lies at distance at most $\le \log(3)$ from any union of two edges), see for instance [Bow06, Section 6.1]) so at each step at least one of the two edges pointing away from the root is at distance $\le \log(3)/2$ from the foot of $c_i$. It follows that $c$ has length at most $(S+1)\log(3)$ and we conclude that $x$ is at distance at most $\log(3) \cdot (S+1)$ from the boundary. From this and (3.2) we deduce that at least $(1 - O(2^{-S}))|T_n|$ triangles of $T_n$ lie entirely in $(P'_n)_{\le R}$, for $S = R/\log(3)$. If $A$ is a lower bound for the area of triangles in $T_n$ (which is independent of $n$ by the remarks above) we thus have that

$$ \text{Area}(P'_n) \ge (1 - O(2^{-R}))A. $$

On the other hand we have that

$$ \text{Area}(P'_n \setminus (P'_n)_{\le R}) \le \pi \cdot 2^{\log_2(n)+1-S} $$

so that

$$ \text{Area}((P'_n)_{\le R}) \ge (1 - O(2^{-R})) \text{Area}(P'_n). $$

from which (3.1) follows immediately.

### 4. CHABAUTY LIMITS OF COXETER POLYGONS

In this section we prove the following result, which immediately implies the second part of Theorem 3 since limits of discrete invariant random subgroups are themselves supported on discrete subgroups (as follows from [ABB$^+$17, Proposition 2.2, Theorem 2.9]).

**Proposition 7.** *The set of discrete groups generated by reflections is closed in the Chabauty space of discrete subgroups of PO(2, 1).*

This will follow from the next lemma.

**Lemma 8.** *Let $P$ be a Coxeter polygon in $\mathbb{H}^2$, $S$ the set of reflections in its faces and $\Gamma = \Gamma_P = \langle S \rangle$. If $w \in \Gamma$ and $T$ is the subset of $S$ containing all elements occuring in a minimal expression for $w$ as a word in the elements of $S$ then $d(x, wx) \ge d(x, sx)$ for all $s \in T$.*

*Proof.* Assume that $s \in S$ occurs in a minimal expression for $w$; then $x$ and $wx$ are separated by an hyperplane which is $\Gamma$-equivalent to the hyperplane $W_s$ supporting the side of $P$ corresponding to $s$. Thus we need only show the following statement: the smallest distance between two points in the orbit $\Gamma \cdot x$ separated by $W_s$ is realised by $(x, sx)$.

In turn this is implied by the statement that the closest point to $x$ on a $\Gamma$-translate of $W_s$ is its projection on $W_s$. If this were not the case there would be a billiard trajectory in $P$ starting at $x$ and ending on $W_s$ shorter than the segment from $x$ orthogonal to $W_s$. This is impossible: indeed, there is no trajectory from $x$ to $W_s$ at all which is shorter than this segment. This finishes the proof. □
$(T_n)_{\le S}$ and the edges $(t_i, t_{i+1})$ point away from the root we must have that $l \le S + 1$. Moreover, the length of each $c_i$ is at most $\log(3)$ since $\mathbb{H}^2$ is $\log(3)/2$-hyperbolic (which implies that in any geodesic triangle any point lies at distance at most $\le \log(3)$ from any union of two edges), see for instance [Bow06, Section 6.1]) so at each step at least one of the two edges pointing away from the root is at distance $\le \log(3)/2$ from the foot of $c_i$. It follows that $c$ has length at most $(S+1)\log(3)$ and we conclude that $x$ is at distance at most $\log(3) \cdot (S+1)$ from the boundary. From this and (3.2) we deduce that at least $(1-O(2^{-S}))|T_n|$ triangles of $T_n$ lie entirely in $(P'_n)_{\le R}$, for $S=R/\log(3)$. If $A$ is a lower bound for the area of triangles in $T_n$ (which is independent of $n$ by the remarks above) we thus have that

$$ \text{Area}(P'_n) \ge (1 - O(2^{-R}))A. $$

On the other hand we have that

$$ \text{Area}(P'_n \setminus (P'_n)_{\le R}) \le \pi \cdot 2^{\log_2(n)+1-S} $$

so that

$$ \text{Area}((P'_n)_{\le R}) \ge (1 - O(2^{-R})) \text{Area}(P'_n). $$

from which (3.1) follows immediately.

### 4. CHABAUTY LIMITS OF COXETER POLYGONS

In this section we prove the following result, which immediately implies the second part of Theorem 3 since limits of discrete invariant random subgroups are themselves supported on discrete subgroups (as follows from [ABB$^+$17, Proposition 2.2, Theorem 2.9]).

**Proposition 7.** The set of discrete groups generated by reflections is closed in the Chabauty space of discrete subgroups of $\text{PO}(2, 1)$.

This will follow from the next lemma.

**Lemma 8.** Let $P$ be a Coxeter polygon in $\mathbb{H}^2$, $S$ the set of reflections in its faces and $\Gamma = \Gamma_P = \langle S \rangle$. If $w \in \Gamma$ and $T$ is the subset of $S$ containing all elements occuring in a minimal expression for $w$ as a word in the elements of $S$ then $d(x, wx) \ge d(x, sx)$ for all $s \in T$.

*Proof.* Assume that $s \in S$ occurs in a minimal expression for $w$; then $x$ and $wx$ are separated by an hyperplane which is $\Gamma$-equivalent to the hyperplane $W_s$ supporting the side of $P$ corresponding to $s$. Thus we need only show the following statement: the smallest distance between two points in the orbit $\Gamma \cdot x$ separated by $W_s$ is realised by $(x, sx)$.

In turn this is implied by the statement that the closest point to $x$ on a $\Gamma$-translate of $W_s$ is its projection on $W_s$. If this were not the case there would be a billiard trajectory in $P$ starting at $x$ and ending on $W_s$ shorter than the segment from $x$ orthogonal to $W_s$. This is impossible: indeed, there is no trajectory from $x$ to $W_s$ at all which is shorter than this segment. This finishes the proof. $\square$
*Proof of Proposition 7.* Let $H$ be a discrete Chabauty limit of a sequence $\Gamma_n$ of Coxeter groups in $\mathbb{H}^2$. Since $H$ is discrete, it follows from the Kazhdan–Margulis theorem that there exists $\varepsilon > 0$ such that, by conjugating the $\Gamma_n$, we may assume there is a point $o \in \mathbb{H}^2$ which maps to the $\varepsilon$-thick part of $\Gamma_n \backslash \mathbb{H}^2$ for every $n$. Let $P_n$ be the Coxeter polygon of $\Gamma_n$ containing $o$.

Fix a $g \in H$: we want to prove that $g$ is a product of reflections in $H$. We know that $g$ is a limit of a sequence $g_n \in \Gamma_n$. The distances $d(o, g_n o)$ are uniformly bounded, say $d(o, g_n o) \le R$ for all $n$, and it follows that the word length of $g_n$ in the reflections in the sides of $P_n$ must be bounded, say by some $l \in \mathbb{N}$ (since there are uniformly finitely many points in the orbit $\Gamma_n \cdot o$ at distance at most $R$ from $o$ by a packing argument). By Lemma 8 it follows that we have $g_n = s_{i_1,n} \cdots s_{i_n,n}$ where $s_{i,n}$ are reflections in the sides of $P_n$ such that $d(o, s_{i,n}o) \le R$ and $l_n \le l$ for all $n$. We can thus pass to a subsequence and assume that the sequences $s_{i,n}$, $1 \le i \le l$ are convergent. The limits are reflections belonging to $H$ and $g$ can be written as a product of them. $\square$

## REFERENCES

[ABB$^{+}$17] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, and Iddo Samet. On the growth of $L^2$-invariants for sequences of lattices in Lie groups. *Ann. Math. (2)*, 185(3):711–790, 2017.

[ABSW08] Ian Agol, Mikhail Belolipetsky, Peter Storm, and Kevin Whyte. Finiteness of arithmetic hyperbolic reflection groups. *Groups Geom. Dyn.*, 2(4):481–498, 2008.

[Bel16] Mikhail Belolipetsky. Arithmetic hyperbolic reflection groups. *Bull. Am. Math. Soc.*, New Ser., 53(3):437–475, 2016.

[Bow06] Brian H. Bowditch. *A course on geometric group theory*, volume 16 of MSJ Mem. Tokyo: Mathematical Society of Japan (MSJ), 2006.

[FH22] David Fisher and Sebastian Hurtado. A new proof of finiteness of maximal arithmetic reflection groups, 2022.

[FHR23] Mikolaj Fraczyk, Sebastian Hurtado, and Jean Raimbault. Benjamini–Schramm convergence of arithmetic locally symmetric spaces. In preparation, see https://arxiv.org/abs/2202.13619, 2023.

[Gro07] Misha Gromov. *Metric structures for Riemannian and non-Riemannian spaces*. *Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes*. Edited by J. LaFontaine and P. Pansu. Mod. Birkhäuser Classics. Basel: Birkhäuser, 3rd printing edition, 2007.

[Kho86] A. G. Khovanskij. Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskij space. *Funct. Anal. Appl.*, 20:41–50, 1986.

[KLM$^{+}$18] Matias Korman, Stefan Langerman, Wolfgang Mulzer, Alexander Pilz, Maria Saumell, and Birgit Vogtenhuber. The dual diameter of triangulations. *Com-put. Geom.*, 68:243–252, 2018.

[Med12] A. D. Mednykh. Brahmagupta formula for cyclic quadraterials in the hyperbolic plane. *Sib. Èlektron. Mat. Izv.*, 9:247–255, 2012.

[Nik07] V. V. Nikulin. Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces. *Izv. Math.*, 71(1):53–56, 2007.

[Vin81] È. B. Vinberg. Absence of crystallographic reflection groups in Lobachevskii spaces of large dimension. *Funkts. Anal. Prilozh.*, 15(2):67–68, 1981.
*Proof of Proposition 7.* Let $H$ be a discrete Chabauty limit of a sequence $\Gamma_n$ of Coxeter groups in $\mathbb{H}^2$. Since $H$ is discrete, it follows from the Kazhdan–Margulis theorem that there exists $\varepsilon > 0$ such that, by conjugating the $\Gamma_n$, we may assume there is a point $o \in \mathbb{H}^2$ which maps to the $\varepsilon$-thick part of $\Gamma_n \backslash \mathbb{H}^2$ for every $n$. Let $P_n$ be the Coxeter polygon of $\Gamma_n$ containing $o$.

Fix a $g \in H$: we want to prove that $g$ is a product of reflections in $H$. We know that $g$ is a limit of a sequence $g_n \in \Gamma_n$. The distances $d(o, g_n o)$ are uniformly bounded, say $d(o, g_n o) \le R$ for all $n$, and it follows that the word length of $g_n$ in the reflections in the sides of $P_n$ must be bounded, say by some $l \in \mathbb{N}$ (since there are uniformly finitely many points in the orbit $\Gamma_n \cdot o$ at distance at most $R$ from $o$ by a packing argument). By Lemma 8 it follows that we have $g_n = s_{i_1,n} \cdots s_{i_n,n}$ where $s_{i,n}$ are reflections in the sides of $P_n$ such that $d(o, s_{i,n}o) \le R$ and $l_n \le l$ for all $n$. We can thus pass to a subsequence and assume that the sequences $s_{i,n}$, $1 \le i \le l$ are convergent. The limits are reflections belonging to $H$ and $g$ can be written as a product of them. $\square$

## REFERENCES

[ABB$^{+}$17] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, and Iddo Samet. On the growth of $L^2$-invariants for sequences of lattices in Lie groups. *Ann. Math. (2)*, 185(3):711–790, 2017.

[ABSW08] Ian Agol, Mikhail Belolipetsky, Peter Storm, and Kevin Whyte. Finiteness of arithmetic hyperbolic reflection groups. *Groups Geom. Dyn.*, 2(4):481–498, 2008.

[Bel16] Mikhail Belolipetsky. Arithmetic hyperbolic reflection groups. *Bull. Am. Math. Soc.*, New Ser., 53(3):437–475, 2016.

[Bow06] Brian H. Bowditch. *A course on geometric group theory*, volume 16 of *MSJ Mem*. Tokyo: Mathematical Society of Japan (MSJ), 2006.

[FH22] David Fisher and Sebastian Hurtado. A new proof of finiteness of maximal arithmetic reflection groups, 2022.

[FHR23] Mikolaj Fraczyk, Sebastian Hurtado, and Jean Raimbault. Benjamini–Schramm convergence of arithmetic locally symmetric spaces. In preparation, see https://arxiv.org/abs/2202.13619, 2023.

[Gro07] Misha Gromov. *Metric structures for Riemannian and non-Riemannian spaces*. *Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes*. Edited by J. LaFontaine and P. Pansu. Mod. Birkhäuser Classics. Basel: Birkhäuser, 3rd printing edition, 2007.

[Kho86] A. G. Khovanskij. Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskij space. *Funct. Anal. Appl.*, 20:41–50, 1986.

[KLM$^{+}$18] Matias Korman, Stefan Langerman, Wolfgang Mulzer, Alexander Pilz, Maria Saumell, and Birgit Vogtenhuber. The dual diameter of triangulations. *Com-put. Geom.*, 68:243–252, 2018.

[Med12] A. D. Mednykh. Brahmagupta formula for cyclic quadraterials in the hyperbolic plane. *Sib. Èlektron. Mat. Izv.*, 9:247–255, 2012.

[Nik07] V. V. Nikulin. Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces. *Izv. Math.*, 71(1):53–56, 2007.

[Vin81] È. B. Vinberg. Absence of crystallographic reflection groups in Lobachevskii spaces of large dimension. *Funkts. Anal. Prilozh.*, 15(2):67–68, 1981.
[Vin85] È. B. Vinberg. Hyperbolic reflection groups. *Russ. Math. Surv.*, 40(1):31–75, 1985.

INSTITUT DE MATHÉMATIQUES DE MARSEILLE, UMR 7373, CNRS, AIX-MARSEILLE
UNIVERSITÉ

Email address: jean.raimbault@univ-amu.fr
[Vin85] È. B. Vinberg. Hyperbolic reflection groups. *Russ. Math. Surv.*, 40(1):31–75, 1985.

INSTITUT DE MATHÉMATIQUES DE MARSEILLE, UMR 7373, CNRS, AIX-MARSEILLE
UNIVERSITÉ

Email address: jean.raimbault@univ-amu.fr