Švarc–Milnor lemma

Several minor variations of the statement of the lemma exist in the literature (see the Notes section below). Here we follow the version given in the book of Bridson and Haefliger (see Proposition 8.19 on p. 140 there).[5]

Let ${\displaystyle G}$ be a group acting by isometries on a proper length space ${\displaystyle X}$ such that the action is properly discontinuous and cocompact.

Then the group ${\displaystyle G}$ is finitely generated and for every finite generating set ${\displaystyle S}$ of ${\displaystyle G}$ and every point ${\displaystyle p\in X}$ the orbit map

${\displaystyle f_{p}:(G,d_{S})\to X,\quad g\mapsto gp}$

is a quasi-isometry.

Here ${\displaystyle d_{S}}$ is the word metric on ${\displaystyle G}$ corresponding to ${\displaystyle S}$.

In many sources the Švarc–Milnor lemma is stated under a slightly more restrictive assumption that the space ${\displaystyle X}$ be a geodesic metric space (rather that a length space), and most applications concern this context.

Sometimes a properly discontinuous cocompact isometric action of a group ${\displaystyle G}$ on a proper geodesic metric space ${\displaystyle X}$ is called a geometric action.[6]

Recall that a metric ${\displaystyle X}$ space 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

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

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

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

For Examples 1 through 5 below see pp. 89–90 in the book of de la Harpe.[3] Example 6 is the starting point of the part of the paper of Richard Schwartz.[7]

1. For every ${\displaystyle n\geq 1}$ the group ${\displaystyle \mathbb {Z} ^{n}}$ is quasi-isometric to the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$.

2. If ${\displaystyle \Sigma }$ is a closed connected oriented surface of negative Euler characteristic then the fundamental group ${\displaystyle \pi _{1}(\Sigma )}$ 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 ${\displaystyle \Gamma \leq G}$ is a uniform lattice then ${\displaystyle \Gamma }$ 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 ${\displaystyle \Gamma =\pi _{1}(M)}$ is quasi-isometric to ${\displaystyle \Omega =\mathbb {H} ^{3}-{\mathcal {B}}}$, where ${\displaystyle {\mathcal {B}}}$ is a certain ${\displaystyle \Gamma }$-invariant collection of horoballs, and where ${\displaystyle \Omega }$ is equipped with the induced path metric.