Groups acting non-properly cocompactly on hyperbolic spaces

Question

A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ generated by a finite set $X$ is hyperbolic if the Cayley graph $\Gamma(G;X)$ is a hyperbolic metric space. The equivalence of both the definitions can be provided by Svarc-Milnor Lemma.

My question is:

What are the examples of group (finitely generated and infinite) acting on a geodesic hyperbolic space cocompactly and fails to be hyperbolic? i.e., the action is not proper.

Answer 1

Bass-Serre theory is the classical theory of groups acting on trees. Trees are hyperbolic, so this theory fits nicely into the context of this question.

Standard examples of groups acting on trees co-finitely include free products (possibly with amalgamation) and HNN extensions. Specific exotic/non-hyperbolic examples include $\operatorname{SL}_2$, Baumslag-Solitar groups (the standard examples of finitely presented non-hyperbolic groups), one-relator groups, and the Boone-Britton example of a finitely presented group with undecidable word problem.

Every non-trivial group acts non-trivially on a tree - take the rooted tree with a single root vertex, and children labelled by the group elements; the action is by left multiplication on the labels. Bass-Serre theory therefore assumes that an action does not have a global fixed point. The actions in the above paragraph do not have global fixed points. A group has Serre's property FA if every action on a tree has a global fixed point; such groups include finite groups, triangle groups, and groups with Kazhdan's property (T).

The standard reference is Serre's 1980 book Trees, translated into English by John Stillwell. Meier's book "Groups, Graphs and Trees" offers a more gentle introduction.

Answer 2

As already said in the comments, every group acts cocompactly on a single point, which is hyperbolic. But of course, this is not an satisfying example. Another source of not so interesting actions comes from combinatorial horoballs. If $G$ is a finitely generated group, let it act on one its Cayley graph $X$ (associated to a finite generating set). The combinatorial horoball $\mathcal{H}(X)$ over $X$ is the graph obtained from $V(X) \times \mathbb{N}$ where two vertices $(u,n)$ and $(v,m)$ are connected by an edge if

• $u=v$ and $m=n+1$;
• $n=m$ and $d_X(u,v) \leq 2^n$.

Then $\mathcal{H}(X)$ is a locally finite hyperbolic graph on which $G$ acts with unbounded orbits (if $G$ is infinite). Thus, every finitely generated group acts properly on a hyperbolic space. But the action is not cobounded.

Admitting a cobounded action on an unbounded hyperbolic space is rather common. Groups that do not admit such actions are investigated in arxiv:2212.14292, under the name Property (NL). Constructing examples of such groups is not so obvious.

For cocompact actions, the picture is less clear (at least for me). Of course, groups with infinite hyperbolic groups are examples, and they may be far from being hyperbolic. This includes finitely generated groups with infinite abelianisations. For an explicit family of examples, consider for instance right-angled Artin groups. Namely, given a graph $\Gamma$, define $$A(\Gamma):= \langle u \in V(\Gamma) \mid [u,v]=1, \ \{u,v\} \in E(\Gamma) \rangle.$$ If $\Gamma$ is not a complete graph, i.e. there are two non-adjacent vertices $u,v \in V(\Gamma)$, then $A(\Gamma)$ retracts naturally onto $\langle u,v \rangle \simeq \mathbb{F}_2$. On the other hand, $A(\Gamma)$ is hyperbolic if and only if $\Gamma$ has no edge.

Again, such examples are not entirely satisfying, because the actions have huge kernels. As mentioned by ADL, amalgamated products and HNN extensions (or more generally, fundamental groups of finite graphs of groups) are examples of groups acting cocompactly on trees, but they may be far different from hyperbolic groups (e.g. Baumslag-Solitar groups). In the same vein, fundamental groups of complexes of groups can be considered. Examples I like are Higman groups: for every $n \geq 4$, $$H_n:= \langle a_i, \ i \in \mathbb{Z}/n \mathbb{Z} \mid a_ia_{i+1}a_i^{-1}=a_{i+1}^2 \rangle.$$ These groups can be described as simple polygons of groups (with vertex-groups isomorphic to $BS(1,2)$, infinite cyclic edge-groups, and a trivial face-group). The complex of groups is developpable, so $H_n$ naturally acts on a two-dimensional polygonal complex. For $n \geq 5$, this complex is hyperbolic.

Conclusion of the story: many groups act cocompactly on unbounded hyperbolic spaces, and even more when cocompactly is replaced with coboundedly*.

(*It would be interesting to have explicit examples of finitely generated groups acting coboundedly on unbounded hyperbolic spaces but not admiting a cocompact action. It probably exists, but I do not have a simple example in mind right now.)

Answer 3

If $X$ is a manifold with a complete geodesic word-hyperbolic metric, and the action is cocompact and indiscrete, then Montgomery-Zippin should imply that $X$ is a rank one symmetric space and $G$ is a dense subgroup (like the examples given by Yves de Cornulier in the comments). If $G$ is not discrete, then its closure in $Isom(X)$ should be a Lie group with a transitive action by Montgomery-Zippin.

More generally, one can take dense subgroups of a locally compact hyperbolic topological group, such as the isometry group of a regular finite-valence tree or examples in this paper.

As indicated in the comments, without the assumption of local compactness or faithfulness, any finitely generated group has such an action (on a hyperbolic space with cocompact action).

Answer 4

Several interesting (and useful) examples of faithful, cocompact, and nonproper actions of groups on hyperbolic spaces occur under the umbrella of acylindrically hyperbolic groups including hierchically hyperbolic groups, and other related examples. These examples do tend to be non-locally compact; nonetheless they are cocompact by virtue of having nice compact subsets whose translates cover the whole space. For instance:

• The action of the mapping class group $\text{MCG}(S)$ of a finite type surface $S$ on the curve complex of $S$ (Masur and Minsky);
• The action of virtually compact special groups on certain quasitrees called contact graphs (Behrstock, Hagen and Sisto);
• The actions of the outer automorphism group $\text{Out}(F_n)$ of a finite rank free group $F_n$ on its free factor complex (Bestvina and Feighn) and on its free splitting complex (Handel and myself).