In the mathematical subject of geometric group theory, the Švarc–Milnor lemma is a statement which says that a group, equipped with a "nice" discrete isometric action on a metric space, is quasi-isometric to. This result goes back, in different form, before the notion of quasi-isometry was formally introduced, to the work of Albert S. Schwarz and John Milnor. Pierre de la Harpe called the Švarc–Milnor lemma ``the fundamental observation in geometric group theory" 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.
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. Let be a group acting by isometries on a proper length space such that the action is properly discontinuous and cocompact. Then the group is finitely generated and for every finitegenerating set of and every point the orbit map is a quasi-isometry. Here is the word metric on corresponding to.
Explanation of the terms
Recall that a metric space is proper if every closed ball in is compact. An action of on is properly discontinuous if for every compact the set is finite. The action of on is cocompact if the quotient space, 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 in such that
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 the group is quasi-isometric to the Euclidean space. 2. If is a closed connected oriented surface of negative Euler characteristic then the fundamental group is quasi-isometric to the hyperbolic plane. 3. If is a closed connected smooth manifold with a smooth Riemannian metric then is quasi-isometric to, where is the universal cover of, where is the pull-back of to , and where is the path metric on defined by the Riemannian metric. 4. If is a connected finite-dimensional Lie group equipped with a left-invariant Riemannian metric and the corresponding path metric, and if is a uniform lattice then is quasi-isometric to. 5. If is a closed hyperbolic 3-manifold, then is quasi-isometric to . 6. If is a complete finite volume hyperbolic 3-manifold with cusps, then is quasi-isometric to , where is a certain -invariant collection of horoballs, and where is equipped with the induced path metric.
