Švarc–Milnor lemma

Explanation of the terms

Recall that a metric space ${\displaystyle X}$ is proper if every closed ball in ${\displaystyle X}$ is compact.

An action of ${\displaystyle G}$ on ${\displaystyle X}$ is properly discontinuous if for every compact ${\displaystyle K\subseteq X}$ the set

${\displaystyle \{g\in G\mid gK\cap K\ne \varnothing \}}$

is finite.

The action of ${\displaystyle G}$ on ${\displaystyle X}$ is cocompact if the quotient space ${\displaystyle X/G}$, equipped with the quotient topology, is compact. Under the other assumptions of the Švarc–Milnor lemma, the cocompactness condition is equivalent to the existence of a closed ball ${\displaystyle B}$ in ${\displaystyle X}$ such that

${\displaystyle \bigcup _{g\in G}gB=X.}$

Examples of applications of the Švarc–Milnor lemma

For Examples 1 through 5 below see pp. 89–90 in the book of de la Harpe. Example 6 is the starting point of the part of the paper of Richard Schwartz.

1. For every ${\displaystyle n\ge 1}$ the group ${\displaystyle {\mathbb{Z}}^n}$ is quasi-isometric to the Euclidean space ${\displaystyle {\mathbb{R}}^n}$.
2. If ${\displaystyle \mathrm{\Sigma }}$ is a closed connected oriented surface of negative Euler characteristic then the fundamental group ${\displaystyle {\pi }_1(\mathrm{\Sigma })}$ is quasi-isometric to the hyperbolic plane ${\displaystyle {\mathbb{H}}^2}$.
3. If ${\displaystyle (M,g)}$ is a closed connected smooth manifold with a smooth Riemannian metric ${\displaystyle g}$ then ${\displaystyle {\pi }_1(M)}$ is quasi-isometric to ${\displaystyle (\tilde{M},d_{\tilde{g}})}$, where ${\displaystyle \tilde{M}}$ is the universal cover of ${\displaystyle M}$, where ${\displaystyle \tilde{g}}$ is the pull-back of ${\displaystyle g}$ to ${\displaystyle \tilde{M}}$, and where ${\displaystyle d_{\tilde{g}}}$ is the path metric on ${\displaystyle \tilde{M}}$ defined by the Riemannian metric ${\displaystyle \tilde{g}}$.
4. If ${\displaystyle G}$ is a connected finite-dimensional Lie group equipped with a left-invariant Riemannian metric and the corresponding path metric, and if ${\displaystyle \mathrm{\Gamma }\le G}$ is a uniform lattice then ${\displaystyle \mathrm{\Gamma }}$ is quasi-isometric to ${\displaystyle G}$.
5. If ${\displaystyle M}$ is a closed hyperbolic 3-manifold, then ${\displaystyle {\pi }_1(M)}$ is quasi-isometric to ${\displaystyle {\mathbb{H}}^3}$.
6. If ${\displaystyle M}$ is a complete finite volume hyperbolic 3-manifold with cusps, then ${\displaystyle \mathrm{\Gamma }={\pi }_1(M)}$ is quasi-isometric to ${\displaystyle \mathrm{\Omega }={\mathbb{H}}^3-\mathcal{B}}$, where ${\displaystyle \mathcal{B}}$ is a certain ${\displaystyle \mathrm{\Gamma }}$-invariant collection of horoballs, and where ${\displaystyle \mathrm{\Omega }}$ is equipped with the induced path metric.