Švarc–Milnor lemma

In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group ${\displaystyle G}$, equipped with a "nice" discrete isometric action on a metric space ${\displaystyle X}$, is quasi-isometric to ${\displaystyle X}$.

This result goes back, in different form, before the notion of quasi-isometry was formally introduced, to the work of Albert S. Schwarz (1955) ^[1] and John Milnor (1968). ^[2] Pierre de la Harpe called the Švarc–Milnor lemma "the fundamental observation in geometric group theory" ^[3] because of its importance for the subject. Occasionally the name "fundamental observation in geometric group theory" is now used for this statement, instead of calling it the Švarc–Milnor lemma; see, for example, Theorem 8.2 in the book of Farb and Margalit. ^[4]

Precise statement

Several minor variations of the statement of the lemma exist in the literature. 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 ${\displaystyle S}$.

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]

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

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

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

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

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. ^[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.

References

1. A. S. Švarc, A volume invariant of coverings(in Russian), Doklady Akademii Nauk SSSR, vol. 105, 1955, pp. 32–34.
2. J. Milnor, A note on curvature and fundamental group, Journal of Differential Geometry, vol. 2, 1968, pp. 1–7
3. Pierre de la Harpe, Topics in geometric group theory . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN   0-226-31719-6; p. 87
4. Benson Farb, and Dan Margalit, A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012. ISBN   978-0-691-14794-9; p. 224
5. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ISBN   3-540-64324-9
6. I. Kapovich, and N. Benakli, Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, American Mathematical Society, Providence, RI, 2002, ISBN   0-8218-2822-3; Convention 2.22 on p. 46
7. Richard Schwartz, The quasi-isometry classification of rank one lattices, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, vol. 82, 1995, pp. 133–168