Hyperbolic Groups

Introduction

A metric space is hyperbolic in the sense of Gromov if its geodesic triangles are uniformly thin: there is a constant $\delta \geq 0$ such that every side of every geodesic triangle lies in the $\delta$-neighbourhood of the union of the other two. Trees are $0$-hyperbolic, hyperbolic space $\mathbf{H}^n$ is hyperbolic for some $\delta$ depending on $n$, and the condition is a coarse one: it is invariant under quasi-isometry. A finitely generated group is hyperbolic when its Cayley graph is hyperbolic; the class contains the free groups, the fundamental groups of closed negatively curved manifolds, the uniform lattices in rank-one groups, and a large family of groups defined by presentations, and it excludes $\mathbb{Z}^2$, all groups containing it, and every group with a Baumslag–Solitar subgroup generated by a pair acting with a large discrepancy.

The theory is one of the most complete in geometric group theory. Hyperbolic spaces have a well-behaved theory of quasi-geodesics — the Morse lemma says that a quasi-geodesic stays close to a geodesic — and a boundary at infinity, the space of geodesic rays up to bounded distance, which is compact for a proper space and carries a visual family of metrics. A quasi-isometry of proper hyperbolic spaces extends to a homeomorphism of the boundaries, so the boundary is a quasi-isometry invariant. For a hyperbolic group this yields a powerful set of consequences: the group is finitely presented, the word problem is solvable by Dehn's algorithm in linear time, the Rips complex gives a finite model for the classifying space, and a non-elementary hyperbolic group contains a free group of rank two and has one end and exponential growth. The combination of the boundary with the local-to-global structure also gives the rigidity theory: quasi-isometric rigidity of hyperbolic groups, and Cannon's conjecture, which asks whether a hyperbolic group whose boundary is a two-sphere acts geometrically on hyperbolic three-space.

The article develops the definition and the first examples, the Morse lemma, the boundary at infinity and its quasi-isometry invariance, the theory of hyperbolic groups with the classification of isometries and the Rips complex, the algorithmic properties, and the rigidity statements. The metric input is that of Geometric Group Theory, immediately above: the Cayley graph, the word metric, quasi-isometry and the growth of groups. The geometric input on hyperbolic space is that of Lattices in Lie Groups, where $\mathbf{H}^n$ appears as the homogeneous space $SO(n,1)/SO(n)$ with a proper distance; the boundary of $\mathbf{H}^n$ is the sphere $S^{n-1}$. The action of a group on a tree, which is the degenerate case $\delta = 0$, is not covered here.

The boundary with Part III is the one fixed for this block. What is developed here is the metric, topological and combinatorial structure: the thin-triangle condition, quasi-geodesics, the boundary with its topology, the classification of isometries, the Rips complex and the algorithmic properties. What is deferred is the analytic theory: the Patterson–Sullivan and quasi-conformal measures on the boundary, the Hausdorff dimension of the limit sets, the ergodic theory of the geodesic flow and the spectral theory of the Laplacian on a hyperbolic quotient, all of which belong to Analysis on Groups, where the measure and the limit are available. No physics is invoked.

Gromov Hyperbolic Spaces

The Thin-Triangle Condition

Definition. Let $X$ be a metric space and let $x,y,z \in X$. A geodesic triangle with vertices $x,y,z$ consists of three geodesic segments joining the vertices in pairs. For $\delta \geq 0$ the triangle is $\delta$-thin if each of its sides lies in the $\delta$-neighbourhood of the union of the other two. The space $X$ is $\delta$-hyperbolic if it is geodesic and every geodesic triangle is $\delta$-thin, and hyperbolic if it is $\delta$-hyperbolic for some $\delta$. The number $\delta$ is the hyperbolicity constant; a $0$-hyperbolic space is one whose triangles are degenerate, and a tree is the model of this case.

Theorem (equivalence of the definitions). For a geodesic metric space $X$ the following conditions are equivalent for suitable constants $\delta_1, \delta_2$ depending only on each other:

(a) every geodesic triangle is $\delta_1$-thin (the thin-triangle condition);

(b) for all $p, x, y, z \in X$ the two larger of the three numbers $d(p,x)+d(y,z)$, $d(p,y)+d(x,z)$, $d(p,z)+d(x,y)$ differ by at most $2\delta_2$ (the four-point condition);

(c) for all $p, x, y, z$, the distance from $p$ to the geodesic $[xy]$ is at most $\delta_2 + \max\{d(p,[xz]), d(p,[yz])\}$ (the Gromov product condition).

Proof sketch. The four-point condition is a rearrangement of the thin-triangle condition: a tripod comparison of the four points, in which one collapses the pairs $(y,z),(x,z),(x,y)$ to a tripod with centre $o$, shows that the excess in (b) equals twice the distance from $p$ to the union of the three sides, which by (a) is at most $\delta_1$; the converse reverses the comparison. Condition (c) is the same inequality written in terms of the Gromov products $(x\vert y)_p = \tfrac12(d(p,x)+d(p,y)-d(x,y))$, and the two constants differ by a bounded amount. The equivalence is standard and is quoted from the literature.

Definition. For $p \in X$ the Gromov product of $x, y$ with respect to $p$ is

$$ (x\vert y)_p = \tfrac{1}{2}\bigl(d(p,x)+d(p,y)-d(x,y)\bigr) , $$

a non-negative quantity measuring how long the geodesics from $p$ to $x$ and from $p$ to $y$ travel together. In a tree the product is the length of the common initial segment; in general $X$ is $\delta$-hyperbolic exactly when the products satisfy the four-point inequality up to $2\delta$, which is the form of hyperbolicity used in the construction of the boundary.

Examples and Non-Examples

Example (trees). A tree is $0$-hyperbolic. Every geodesic triangle in a tree is a tripod: the three sides meet at a single point, and each side is contained in the union of the other two without any thickening. A finitely generated group acting on a tree with a cocompact action — for instance a free group with its Cayley tree — therefore has a $0$-hyperbolic Cayley graph.

Example (hyperbolic space). The hyperbolic space $\mathbf{H}^n$ of Lattices in Lie Groups, the homogeneous space $SO(n,1)/SO(n)$ with a $G$-invariant proper distance, is $\delta$-hyperbolic for a constant depending only on $n$: the thinness of the triangles is the standard property of negatively curved geometry, and it is the reason the large-scale geometry of $\mathbf{H}^n$ is tree-like. The boundary at infinity is the sphere $S^{n-1}$, and the geodesic rays are the half-lines asymptotic to points of the sphere.

Example (the flat plane). The Euclidean space $\mathbb{R}^n$ is not hyperbolic for $n \geq 2$: the triangles of an $n$-dimensional grid contain arbitrarily large Euclidean balls, and the four-point condition fails with an excess growing like the size of the triangle. Consequently no group quasi-isometric to $\mathbb{R}^n$ with $n \geq 2$ is hyperbolic, and in particular $\mathbb{Z}^n$ with $n \geq 2$ is not hyperbolic.

Example (a quasi-isometric image). Hyperbolicity is invariant under quasi-isometry: if $X$ and $Y$ are quasi-isometric geodesic spaces and $X$ is $\delta$-hyperbolic then $Y$ is $\delta'$-hyperbolic with $\delta'$ depending on $\delta$ and on the constants of the quasi-isometry. The invariance is proved by the four-point condition, which is a statement about the distortion of quadruples of points; it is the analogue for hyperbolicity of the invariance of the growth and the ends established in Geometric Group Theory.

Quasi-Geodesics and the Morse Lemma

Definition. Let $I \subseteq \mathbb{R}$ be an interval. A map $c : I \to X$ is a $(\lambda, C)$-quasi-geodesic if

$$ \tfrac{1}{\lambda}|t - t'| - C \leq d(c(t), c(t')) \leq \lambda|t-t'| + C \qquad \text{for all } t,t' \in I, $$

so that a quasi-geodesic is a quasi-isometric embedding of an interval; it is a geodesic when $\lambda = 1$ and $C = 0$. The Morse lemma is the assertion that in a hyperbolic space quasi-geodesics do not stray far from geodesics.

Theorem (Morse lemma). Let $X$ be a $\delta$-hyperbolic geodesic space and let $\lambda \geq 1$, $C \geq 0$. There is a constant $R = R(\delta, \lambda, C)$ such that every $(\lambda,C)$-quasi-geodesic $c$ with endpoints $x,y$ lies in the $R$-neighbourhood of every geodesic $[xy]$, and conversely every geodesic $[xy]$ lies in the $R$-neighbourhood of $c$.

Proof sketch. The proof is by a local-to-global argument. One shows first that a quasi-geodesic is uniformly close to a continuous quasi-geodesic with the same constants, and that a continuous quasi-geodesic can be replaced by a path whose successive points are close; the thin-triangle condition then forces the path to be trapped in a bounded neighbourhood of the geodesic, because each excursion away from the geodesic can be shortened by projecting back and the hyperbolicity bounds the shortening by a quantity depending only on $\delta$. The standard proof is due to Gromov; the metric statement is quoted from the literature.

Corollary (stability). In a hyperbolic space the image of a quasi-isometric embedding of $\mathbb{R}$ is at bounded distance from a geodesic, and the geodesic realisation of a quasi-isometric embedding is unique up to bounded distance: if two geodesics have the same endpoints (or the same two ends at infinity) then they are at Hausdorff distance at most $2\delta$.

Proof. The first statement is the Morse lemma applied to the quasi-geodesic on a compact interval and then to the exhaustion of $\mathbb{R}$; the second is the case $\lambda = 1$, $C = 0$ of the stability statement, where the two geodesics, being quasi-geodesics with the same endpoints, lie in each other's $R$-neighbourhood, and the same argument applied to the subsegments gives the bound $2\delta$.

Remark. The Morse lemma fails in $\mathbb{R}^2$: a polygonal line of small steps from $(0,0)$ to $(n,0)$ is a quasi-geodesic with uniform constants, and its distance from the straight segment is unbounded as $n$ grows. The failure of the lemma is equivalent to the failure of hyperbolicity, and it is the reason hyperbolic spaces have a well-defined boundary: the quasi-geodesics and the geodesics define the same ends, so a quasi-isometry can be transported to the boundary.

The Boundary at Infinity

Geodesic Rays and the Boundary

Definition. Let $X$ be a proper hyperbolic geodesic space. Two geodesic rays $c, c' : [0,\infty) \to X$ are asymptotic if $\sup_t d(c(t), c'(t)) < \infty$; this is an equivalence relation. The boundary at infinity $\partial X$ is the set of asymptotic classes of geodesic rays, with the topology for which a sequence of rays converges when the rays converge uniformly on compact sets after passing to suitable parametrisations, and with a base point $p \in X$ one writes $[\gamma]$ for the class of the ray from $p$ to a point of the boundary.

Theorem (the boundary). Let $X$ be a proper hyperbolic geodesic space.

(a) Every geodesic ray from a base point $p$ extends to a unique point of $\partial X$, and every point of $\partial X$ is represented by a unique geodesic ray issuing from $p$; thus $\partial X$ is in bijection with the set of geodesic rays from $p$.

(b) $\partial X$ is compact and, when $X$ is non-elementary, perfect; the space $\overline{X} = X \cup \partial X$ with the topology generated by the sets of points and the shadow sets of geodesic rays is compact and contains $X$ as a dense open subset.

(c) If $X$ is a tree, $\partial X$ is the space of ends of the tree with the Cantor-set topology; if $X = \mathbf{H}^n$, $\partial X = S^{n-1}$.

(d) A quasi-isometry of proper hyperbolic spaces $f : X \to Y$ induces a homeomorphism $\partial f : \partial X \to \partial Y$, which is independent of the choice of the quasi-isometry up to a bounded indeterminacy; hence the homeomorphism type of $\partial X$ is a quasi-isometry invariant.

Proof sketch. (a) A ray from $p$ stays in the $\delta$-neighbourhood of the triangle formed by $p$ and two points on the ray, so rays diverge at a bounded rate and the asymptotic classes are the same as the classes of rays from $p$; the existence of a ray in each class follows from properness and an Arzelà–Ascoli argument on the compact exhaustion of $X$. (b) Compactness of $\overline{X}$ reduces to the compactness of the shadow closure; perfectness fails only when $\partial X$ has at most two points, which happens exactly for the quasi-isometric images of $\mathbb{R}$. (c) The two identifications are standard computations. (d) A quasi-isometry carries geodesic rays to quasi-geodesic rays, the Morse lemma attaches to each a genuine geodesic ray at bounded distance, and two quasi-geodesics at bounded distance give the same boundary point; the construction is continuous by the stability of quasi-geodesics and invertible by applying it to the quasi-inverse. The details are standard and are quoted from the literature.

Corollary. Hyperbolicity of a group is a quasi-isometry invariant, and the boundary of a hyperbolic group is well defined up to homeomorphism: for a finitely generated group $\Gamma$ with finite generating sets $S$ and $T$, the boundaries $\partial\operatorname{Cay}(\Gamma,S)$ and $\partial\operatorname{Cay}(\Gamma,T)$ are canonically homeomorphic, so one writes $\partial\Gamma$. Up to homeomorphism, $\partial\Gamma$ is an invariant of the group.

Proof. The identity map between the two Cayley graphs is a quasi-isometry by the bi-Lipschitz comparison of the word metrics of Geometric Group Theory; the previous theorem gives the homeomorphism, and the invariance of hyperbolicity under quasi-isometry gives the first statement.

Visual Metrics

Definition. Let $X$ be a proper hyperbolic space with base point $p$ and let $\varepsilon > 0$ be small. For $\xi \neq \eta \in \partial X$ let

$$ \rho_\varepsilon(\xi,\eta) = e^{-\varepsilon(\xi\vert\eta)_p} , $$

where $(\xi\vert\eta)_p$ is the Gromov product of two representatives. For $\varepsilon$ sufficiently small, $(\xi,\eta) \mapsto \rho_\varepsilon(\xi,\eta)$ is comparable to a metric on $\partial X$ that induces the boundary topology; the resulting metrics are the visual metrics, and they are pairwise Hölder equivalent for different choices of $\varepsilon$ and of the base point.

Theorem (quasi-symmetry). A quasi-isometry $f : X \to Y$ of proper hyperbolic spaces induces a homeomorphism of boundaries which is quasi-symmetric with respect to visual metrics: there is a control function $\eta : [0,\infty) \to [0,\infty)$ with $\eta(t) \to 0$ as $t \to 0$ such that

$$ \frac{d(\xi_1,\xi_2)}{d(\xi_1,\xi_3)} < t \;\Longrightarrow\; \frac{d(f\xi_1,f\xi_2)}{d(f\xi_1,f\xi_3)} < \eta(t) $$

for all triples of distinct boundary points. The quasi-symmetric structure of the boundary is therefore a quasi-isometry invariant finer than the topology, and it is the invariant used in the rigidity theorems.

Proof sketch. The induced boundary map carries the Gromov product of a triple to a Gromov product bounded above and below by affine functions of the original, by the quasi-isometry inequalities applied to a configuration of three geodesic rays; substituting into the visual metric gives a distortion controlled by a function of the ratio, which is the definition of quasi-symmetry. The details are standard.

Hyperbolic Groups

Definition and Elementary Properties

Definition. A finitely generated group $\Gamma$ is hyperbolic if for some (equivalently every) finite generating set the Cayley graph $\operatorname{Cay}(\Gamma,S)$ is a hyperbolic metric space. The group is elementary if it is finite or virtually $\mathbb{Z}$; equivalently, if its boundary has at most two points.

Theorem (basic properties). Let $\Gamma$ be a hyperbolic group.

(a) $\Gamma$ is finitely presented.

(b) $\Gamma$ has a Dehn presentation: a finite presentation for which there is a constant $N$ such that every trivial word of length at least $N$ contains, as a subword, more than half of some relator or of a cyclic permutation of a relator; this is the Dehn property of the next section, and it gives Dehn's algorithm for the word problem. Consequently the word problem in $\Gamma$ is solvable in linear time by a deterministic algorithm.

(c) $\Gamma$ is torsion-free or has finitely many conjugacy classes of finite subgroups, and every infinite-order element of $\Gamma$ acts as a loxodromic isometry of the Cayley graph.

(d) If $\Gamma$ is non-elementary, then it has one end, has exponential growth, contains a free subgroup of rank two, and its boundary is uncountable and perfect.

Proof sketch. (a) and (b): the thin-triangle condition shows that if a word in the generators is trivial, then the loop in the Cayley graph can be decomposed into loops shorter than the original by at most a bounded factor; for a group presentation obtained from the finite set of words of length at most $4\delta+2$ that are trivial in $\Gamma$, the decomposition terminates, and this gives both a finite presentation and Dehn's algorithm. (c) the classification of isometries below gives the statement about infinite-order elements; the finite subgroups are conjugates of the stabiliser of a vertex and are finite by properness, with finitely many conjugacy classes by cocompactness. (d) the one-endedness follows from the connectedness of the boundary; the exponential growth from the existence of a free submonoid generated by a suitable pair of loxodromic elements; the free subgroup of rank two from the ping-pong lemma applied to two loxodromic elements with disjoint fixed sets; and the perfectness of the boundary from the density of the orbit of a boundary point under a loxodromic element. The results are Gromov's and are quoted from the literature.

Example (the standard hyperbolic groups). The free groups of finite rank are hyperbolic, with a tree for a Cayley graph and a Cantor set for a boundary. The fundamental group of a closed surface of genus $g \geq 2$ is hyperbolic; the fundamental group of a closed hyperbolic $n$-manifold is hyperbolic with boundary $S^{n-1}$. Uniform lattices in rank-one simple Lie groups are hyperbolic. The group $\mathbb{Z}^2$, every group containing $\mathbb{Z}^2$ as a subgroup, the group $SL_3(\mathbb{Z})$, and every group with property (T) that contains a copy of $\mathbb{Z}^2$ are not hyperbolic: a hyperbolic group contains no subgroup isomorphic to $\mathbb{Z}^2$, since a finitely generated subgroup of a hyperbolic group is quasi-isometrically embedded — its Cayley graph lies at finite Hausdorff distance from the orbit of a base point — and hyperbolicity is inherited by subsets at finite Hausdorff distance, while $\mathbb{Z}^2$ is not hyperbolic.

The Classification of Isometries

Definition. Let $X$ be a proper hyperbolic space and let $\gamma$ be an isometry of $X$. The translation length of $\gamma$ is

$$ \tau(\gamma) = \inf_{x\in X} d(x,\gamma x) . $$

The isometry $\gamma$ is elliptic if some orbit is bounded (equivalently, if $\gamma$ fixes a point when $X$ is a tree or $\mathbf{H}^n$), hyperbolic or loxodromic if $\tau(\gamma) > 0$, and parabolic if it is neither, so that no orbit is bounded and $\tau(\gamma) = 0$.

Theorem (classification). Let $\gamma$ be an isometry of a proper hyperbolic space $X$.

(a) $\gamma$ is elliptic if and only if some (hence every) orbit is bounded; if $X$ is a tree or $\mathbf{H}^n$ then $\gamma$ is elliptic exactly when it has a fixed point.

(b) $\gamma$ is loxodromic if and only if it has an axis: a geodesic $A$ with $\gamma A = A$ and $\gamma$ acting on $A$ by a translation of length $\tau(\gamma) > 0$. A loxodromic isometry has exactly two fixed points in $\partial X$, the endpoints of the axis, and for every $x\in X$ the orbit $\{\gamma^n x\}$ lies at bounded distance from the axis.

(c) $\gamma$ is parabolic if and only if $\tau(\gamma) = 0$ and some orbit is unbounded; a parabolic isometry fixes exactly one point of $\partial X$ and no point of $X$, and its orbits escape to that boundary point.

(d) If a group acts properly discontinuously and cocompactly on $X$, then every element is elliptic or loxodromic: parabolic elements do not occur. In particular, in a hyperbolic group every element of infinite order is loxodromic.

Proof sketch. (a) Boundedness of an orbit forces the existence of a fixed point in the closure of a bounded convex hull, and for a tree or $\mathbf{H}^n$ the fixed point lies in $X$; conversely a fixed point gives a bounded orbit. (b) If $\tau(\gamma) > 0$, a minimising sequence for the displacement produces a point at which the displacement is attained, and the geodesic through that point and its image is invariant and is the axis; the endpoints of the axis are the only fixed boundary points because an isometry with three fixed boundary points fixes a geodesic triangle and hence has positive displacement bounded by its diameter. (c) A parabolic isometry has no fixed point and no positive displacement, so by hyperbolicity it must fix exactly one boundary point; its orbits escape to that point by the four-point condition. (d) A parabolic isometry of a cocompact action would move a compact fundamental domain by an arbitrarily unbounded amount while keeping the displacement arbitrarily small, contradicting proper discontinuity. The classification is standard and is quoted from the literature.

The Rips Complex

Definition. Let $\Gamma$ be a hyperbolic group and let $d \geq 0$. The Rips complex $P_d(\Gamma)$ is the simplicial complex whose vertices are the elements of $\Gamma$ and whose simplices are the finite subsets of $\Gamma$ of diameter at most $d$ with respect to the word metric of a fixed finite generating set.

Theorem (Rips). Let $\Gamma$ be a $\delta$-hyperbolic group.

(a) For every $d \geq 4\delta+2$ the Rips complex $P_d(\Gamma)$ is contractible; for all $d$ the complex is a finite-dimensional and locally finite simplicial complex on which $\Gamma$ acts properly discontinuously and cocompactly.

(b) Consequently $\Gamma$ has a finite model for the classifying space of proper actions: the quotient $P_d(\Gamma)/\Gamma$ is a finite CW-complex with fundamental group $\Gamma$, and when $\Gamma$ is torsion-free the action is free and the quotient is a finite classifying space $B\Gamma$. Hence $\Gamma$ is of type $FP_\infty$, and a torsion-free hyperbolic group has finite cohomological dimension.

(c) $\Gamma$ is finitely presented, and the finite presentation is obtained from the two-skeleton of $P_d(\Gamma)/\Gamma$.

Proof sketch. (a) The contractibility is proved by exhibiting $P_d(\Gamma)$ as the nerve of a cover of the Cayley graph by balls whose intersections are controlled by the hyperbolicity: for $d \geq 4\delta+2$ the finite sets of diameter at most $d$ have the property that all their elements lie in a bounded neighbourhood of a geodesic, and the Helly-type number of the cover is finite by the thin-triangle condition, so the nerve is contractible by the nerve theorem. The action is proper and cocompact by construction. (b) and (c) follow from (a) and the fact that the quotient is finite because the action is cocompact. The proof is Gromov's and is quoted from the literature.

Algorithmic and Rigidity Properties

Dehn's Algorithm

Theorem (Dehn's algorithm). A hyperbolic group $\Gamma$ has a finite presentation $\langle S \mid R\rangle$ with the Dehn property: there is a constant $N$ such that every word $w$ in $S\cup S^{-1}$ that is trivial in $\Gamma$ and has length $|w| \geq N$ contains a subword $u$ of length $|u| > |r|/2$ for some relator $r \in R$ (or a cyclic permutation or inverse of a relator). Consequently the word problem is solvable by repeatedly replacing such a subword $u$ by its complementary shorter word, obtained from $r$, so that the length strictly decreases; the algorithm terminates in a number of steps linear in the length of the input word.

Proof sketch. The Dehn property is the combinatorial translation of the thin-triangle condition: a trivial word $w$ is a closed loop in the Cayley graph, the loop is filled by geodesic triangles, and the hyperbolicity guarantees that a loop of length at least $N$ contains a subpath that runs parallel to a shorter path for more than half of a relator, giving the replacement. The termination is by the strict decrease of length at each step, and the linearity of the running time uses a finite-state precomputation of the possible replacements by a Dehn algorithm. The theorem is Dehn's, in Gromov's hyperbolic setting, and is quoted from the literature.

Corollary. The word problem in a hyperbolic group is decidable in linear time, and the conjugacy problem is solvable; more precisely, hyperbolic groups are biautomatic, so the word and conjugacy problems have solvable algorithmic forms with uniform bounds.

Rigidity

Theorem (quasi-isometric rigidity). Hyperbolicity is a quasi-isometry invariant: a finitely generated group quasi-isometric to a hyperbolic group is hyperbolic. Moreover a quasi-isometry of non-elementary hyperbolic groups induces a quasi-symmetry of their boundaries, and the boundary with its quasi-symmetric structure determines the group up to quasi-isometry; for the fundamental group of a closed hyperbolic manifold the quasi-isometry classification agrees with the commensurability classification, by Mostow rigidity in its quasi-isometric form (Sullivan's theorem that a quasi-conformal homeomorphism of the sphere at infinity is conformal in dimension at least three).

Proof sketch. The boundary with its quasi-symmetric structure is a quasi-isometry invariant, so a quasi-isometry of hyperbolic groups induces a quasi-symmetry of their boundaries; the reconstruction of the group from the boundary uses the fact that the group acts on the boundary with the convergence property and that the limit set is the whole boundary, so the group is recovered from the action. The details for the manifold case use Mostow rigidity and the boundary sphere; the general statement is the standard quasi-isometric rigidity theory and is quoted from the literature.

Theorem (Cannon's conjecture; statement). Let $\Gamma$ be a hyperbolic group whose boundary $\partial\Gamma$ is homeomorphic to the two-sphere $S^2$. Then $\Gamma$ acts properly discontinuously and cocompactly by isometries on hyperbolic three-space $\mathbf{H}^3$, so $\Gamma$ is a uniform lattice in the group of isometries of $\mathbf{H}^3$ and its boundary is the conformal two-sphere at infinity. The conjecture is open in general; the case in which $\Gamma$ is the fundamental group of a closed hyperbolic three-manifold gives the classical examples, and the conjecture has been verified for the hyperbolic groups that are cubulated and virtually compact special, by the work surrounding the resolution of the virtual Haken conjecture.

Proof sketch (of the known implication). If $\Gamma$ acts geometrically on $\mathbf{H}^3$ then the quotient is a closed hyperbolic three-manifold with fundamental group $\Gamma$ and boundary $S^2$, which gives the reverse implication of the conjecture. The forward implication would identify the topological two-sphere at infinity with the conformal sphere at infinity of $\mathbf{H}^3$ and would prove Cannon's conjecture; the combinatorial approach constructs a metric on the boundary from the word metric and shows that the group acts on the resulting space by uniform quasi-conformal maps, and the cubulation hypothesis supplies enough combinatorial structure on the boundary to carry the argument through. The problem is open in general and is quoted from the literature.

The Boundary with Analysis

• The Patterson–Sullivan measures and the quasi-conformal measures on the boundary, the Hausdorff dimension of the limit set of a Kleinian group and the critical exponent of a hyperbolic group are Analysis on Groups, where the measure is available.
• The ergodicity of the geodesic flow on a negatively curved quotient, the spectral theory of the Laplacian, the resonances and the prime geodesic theorem are Part III.
• The harmonic analysis of the boundary, the Martin boundary and the random walks on hyperbolic groups are Part III.
• The classifying spaces and the Baum–Connes conjecture for hyperbolic groups are the operator-algebraic topics in Topology on Linear Algebras.
• What is not deferred: the definition of hyperbolicity and its equivalences, the Morse lemma, the boundary at infinity as a topological space, the visual metrics and quasi-symmetry, the classification of isometries, the Rips complex and the finiteness properties, Dehn's algorithm, and the rigidity statements.

Summary

A geodesic metric space is $\delta$-hyperbolic if every geodesic triangle is $\delta$-thin, equivalently if the four-point condition holds; trees are $0$-hyperbolic and hyperbolic space $\mathbf{H}^n$ is hyperbolic, while $\mathbb{R}^n$ for $n \geq 2$ is not. Hyperbolicity is a quasi-isometry invariant. The Morse lemma states that a $(\lambda,C)$-quasi-geodesic in a $\delta$-hyperbolic space lies in the $R(\delta,\lambda,C)$-neighbourhood of the geodesic with the same endpoints, and the stability of geodesics is the reason quasi-isometries act on the boundary.

The boundary at infinity $\partial X$ of a proper hyperbolic space is the set of asymptotic classes of geodesic rays, a compact space that recovers the ends of a tree and the sphere $S^{n-1}$ for $\mathbf{H}^n$; a quasi-isometry induces a quasi-symmetric homeomorphism of boundaries, so the boundary is a quasi-isometry invariant of a hyperbolic group, and different generating sets give canonically homeomorphic boundaries.

A finitely generated group is hyperbolic when its Cayley graph is. Hyperbolic groups are finitely presented and have a Dehn presentation, so the word problem is linearly solvable; the Rips complex is contractible for large parameters and gives a finite model for the classifying space of proper actions, so the group is of type $FP_\infty$; every infinite-order element is loxodromic, and a non-elementary hyperbolic group has one end, exponential growth, a free subgroup of rank two, and a perfect uncountable boundary. Isometries of a hyperbolic space are elliptic, parabolic or loxodromic according as their translation length is zero with a bounded orbit, zero with an unbounded orbit, or positive; parabolic elements do not occur for a proper cocompact action. Quasi-isometric rigidity holds for hyperbolic groups and their boundary structure, and Cannon's conjecture asks whether a hyperbolic group with boundary $S^2$ acts geometrically on $\mathbf{H}^3$. The measure-theoretic, spectral and ergodic theory of the boundary and the geodesic flow belongs to Part III.

Summary of Notation

Further Reading

• Mikhael Gromov, Hyperbolic groups, in Essays in Group Theory (Springer, 1987), 75–263, for the definition, the first properties and the Rips complex.
• Élie Cartan, Leçons sur la géométrie des espaces de Riemann (Gauthier-Villars, 1928), for the thin-triangle property of hyperbolic space in its classical form.
• Martin R. Bridson and André Haefliger, Metric Spaces of Non-Positive Curvature (Springer, 1999), for the systematic treatment of hyperbolic spaces, quasi-geodesics and the boundary.
• Michel Coornaert, Thomas Delzant and Athanase Papadopoulos, Géométrie et théorie des groupes: les groupes hyperboliques de Gromov (Springer Lecture Notes 1441, 1990), for the details of the Morse lemma and the boundary.
• James W. Cannon, The combinatorial Riemann mapping theorem, Acta Mathematica 173 (1994), 155–234, for the analytic approach to the boundary and the conjecture.
• Ilya Kapovich and Nadia Benakli, Boundaries of hyperbolic groups, in Combinatorial and Geometric Group Theory (American Mathematical Society, 2002), for a survey of the boundary structure.
• Max Dehn, Über unendliche diskontinuierliche Gruppen, Mathematische Annalen 71 (1911), 116–144, for the origin of Dehn's algorithm.
• Frédéric Paulin, Sur les groupes hyperboliques, Astérisque 292 (2004), for the quasi-isometric rigidity and the boundary theory.
• Michael Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory II (Cambridge University Press, 1993), for the quasi-isometric rigidity and the asymptotic cones.
• Brian H. Bowditch, Convergence groups and configuration spaces, in Geometric Group Theory Down Under (de Gruyter, 1999), for the boundary action and the convergence-group methods.