# Examples of hyperbolic groups

What are some other classes of word-hyperbolic groups other than the finite groups, fundamental groups of surfaces with Euler characteristics negative and virtually free groups?

• All introductions to hyperbolic groups I can think of abound of examples. – YCor Aug 6 '19 at 11:04
• Can you tell me some of those texts because so far all the books that I have referred to (papers by Bowditch, the book metric spaces of non-positive curvature, book by Clara Loh, GGT ) cite only these classes of examples. – Temari Aug 6 '19 at 11:10
• The Bridson-Haefliger book certainly mentions more.... it should mention relation to negative curvature somewhere... mention stability under free product. Of course I also think of the Ghys-Harpe book, which is in French. – YCor Aug 6 '19 at 11:12

Below are some sources of hyperbolic groups. Of course, the list is far from being exhaustive.

1. Groups defined by generators and relations:

• Finitely generated free groups, as their Cayley graphs are simplicial trees.
• If $$\varphi$$ is an atoroidal automorphism of a free group $$\mathbb{F}_n$$, then the extension $$\mathbb{F}_n \rtimes_\varphi \mathbb{Z}$$ is hyperbolic.
• One-relator groups with torsion, namely groups admitting a presentation of the form $$\langle x_1, \ldots, x_n \mid r^m=1 \rangle$$ with $$m \geq 2$$, are hyperbolic.

• Coxeter groups not containing $$\mathbb{Z}^2$$ are hyperbolic. (See Theorem 12.6.1 in Davis' book The geometry and topology of Coxeter groups for a better characterisation.) As a nice particular case, right-angled Coxeter groups defined by finite square-free graphs are hyperbolic.

• Generalising hyperbolic right-angled Coxeter groups, graph products of finite groups over square-free graphs are hyperbolic.
• If $$\Gamma$$ is a topological graph which does not contain two disjoint loops, then the braid group $$B_2(\Gamma)$$ is hyperbolic.
• Small cancellation groups C'(1/6) or C'(1/4)-T(4). A very rich source of two-dimensional hyperbolic groups.
• Random groups in Gromov's density model are hyperbolic and non-elementary below a given density.
2. Fundamental groups:

• Fundamental groups of compact negatively curved Riemannian manifolds are hyperbolic.
• Fundamental groups of hyperbolic 3-manifolds. A general construction is to take a hyperbolic 3-manifold of finite volume and to "fill in" its cusps with tori in different ways. The keyword is Dehn fillings.
• Uniform lattices in real, complex and quaternionic hyperbolic spaces are hyperbolic. Arithmetic constructions may lead to explicit examples.
• If $$f$$ is a pseudo-Anosov homeomorphism of a closed surface $$S$$, then $$\pi_1(S) \rtimes_\varphi \mathbb{Z}$$ is hyperbolic, where $$\varphi$$ denotes the automorphism of $$\pi_1(S)$$ induced by $$f$$. (A reference is Otal's monograph Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3; also, a generalisation to free groups of pseudo-Anosov homeomorphisms can be found in Farb and Mosher's article Convex cocompact subgroups of mapping class groups.)
3. Constructions

• Dehn fillings are now defined purely algebraically, and it is possible to create hyperbolic groups by quotienting relatively hyperbolic groups for instance, or even to create exotic hyperbolic groups from classical hyperbolic groups.
• The family of hyperbolic groups is also stable under various operations, including commensurability, some graphs of groups, and some graph products. For instance, free products (as mentioned by Yves in the comments).

I can add a few precise references if some examples interest you. Otherwise, you have some keywords to help you in your research.

• Thank you, much appreciated! I will try finding the sources on the basis of the keywords provided. – Temari Aug 6 '19 at 14:23
• It's not stable under graph product (not even under direct products). – YCor Aug 6 '19 at 22:56
• "Coxeter groups defined by square-free graphs": probably you don't consider the standard graphs (where edges labeled by $2$ are omitted) but omit instead only edges labeled $\infty$? – YCor Aug 6 '19 at 22:58
• Random groups in Gromov's density model are hyperbolic in all densities; below some density rather ensures they're non-elementary. – YCor Aug 6 '19 at 23:00
• In the last point, I didn't mean that the family of hyperbolic groups is stable by all graph products, but only some of them (similarly for graphs of groups). I refer to Meier's article When is a graph product of hyperbolic groups hyperbolic? for a precise characterisation. – AGenevois Aug 7 '19 at 5:50