Švarc-Milnor Lemma

properly discontinuous

The action is properly discontinuous if for every compact subset $K\subset X$ there are only finitely many $g\in G$ such that $g\cdot K\cap K\ne \varnothing$.

cocompact

An action of a group $G$ on a locally compact space $X$ is called cocompact if there exists a compact subset $A\subset X$ such that $X=G\cdot A$. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space $G\setminus X$.

word metric

Let $G$ be a finitely-generated group $G$, and let $A$ be a finite set of generators for $G$. If $\gamma \in G$ is not the identity element, then the length of $\gamma$ is defined as the minimal number of elements of $A\cup A^{-1}$, counted with multiplicity, such that $\gamma$ can be written as a product of these elements. The length of the identity element is defined to be zero.

The word metric $d_A$ on $G$ with respect to $A$ is then defined by the following formula: for all $\gamma$ and ${\gamma }^{\prime }$ in $G,d_A\left(\gamma ,{\gamma }^{\prime }\right)$ is equal to the length of the product ${\gamma }^{\prime -1}\gamma$. The action of $G$ by left translations on the metric space $\left(G,d_A\right)$ is an action by isometries. If $A$ and $B$ are two finite generating sets for $G$, then the identity mapping between the metric spaces $\left(G,d_A\right)$ and $\left(G,d_B\right)$ is a quasi-isometry.

Theorem

Let $G$ be a group acting by isometries on a proper length space $X$ such that the action is properly discontinuous and cocompact. Then the group $G$ is finitely generated and for every finite generating set $S$ of $G$ and every point $p\in X$ the orbit map

$$f_p:\left(G,d_S\right)\to X,g\mapsto gp$$

is a quasi-isometry. Here $d_S$ is the word metric on $G$ corresponding to $S$.