Groups acting properly discontinuous and cocompactly on the hyperbolic plane by isometries.

geometric-group-theoryhyperbolic-groups

I'm looking for a non-elementary hyperbolic group which is quasi isometric to $\mathbb{H}^2$ (and if possible one quasi-isometric to $\mathbb{H}^3$).

I know the group $\text{PSL}(\mathbb{R})$ acts by isometries on $\mathbb{H}^2$ via the maps
$$ z \mapsto \frac{az + b}{cz + d}. $$
Is it then true that $\text{PSL}(\mathbb{Z})$ as a subgroup acts properly discontinuously and cocompactly on $\mathbb{H}^2$ and by consequence (Svarc-Milnor lemma) $\text{PSL}(\mathbb{Z})$ would be the group I'm looking for? At least if this group is a non-elementary hyperbolic group which I'm also not sure of.

Best Answer

The most explicit examples of groups quasi-isometric to the hyperbolic 3-space are certain Coxeter groups. Coxeter groups are best described by their Coxeter diagrams, such as the one below:

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This diagram describes a group with four generators (corresponding to the nodes). The relators are:

  1. Each generator $s_v$ is an involution, $s_v^2=1$.

  2. If two nodes $v, w$ are not connected by an edge, the corresponding generators commute: $s_v s_w=s_w s_v$.

  3. If two nodes $v, w$ are connected by an unlabelled edge, then the generators "braid": $$ (s_vs_w)^3=1 $$

  4. If two nodes $v, w$ are connected by an edge with the label $n$ (in the graph below, $n=5$) then $$ (s_vs_w)^n=1 $$

The Coxeter group described by the diagram above acts isometrically, properly discontinuously and cocompactly on the hyperbolic 3-space (this takes some work to prove, the key is the existence of the corresponding Goursat tetrahedron in ${\mathbb H}^3$). Hence, this Coxeter group is a (necessarily nonelementary hyperbolic group) quasi-isometric to ${\mathbb H}^3$.

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