Švarc–Milnor lemma

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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 , 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 (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

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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   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 finite generating set   of   and every point   the orbit map

 

is a quasi-isometry.

Here   is the word metric on   corresponding to  .

Sometimes a properly discontinuous cocompact isometric action of a group   on a proper geodesic metric space   is called a geometric action.[6]

Explanation of the terms

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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

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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   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.

References

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  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. ^ a b 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