“Švarc–Milnor lemma”的意思、由来-开放百科全书

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

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

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

and every point

Notes

In many sources the Švarc–Milnor lemma is stated under a slightly more restrictive assumption that the space

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

on a proper geodesic metric space

is called a geometric action.^[6]

Explanation of the terms

The action of

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

such that

Examples of applications of the Švarc–Milnor lemma

Example 6 is the starting point of the part of the paper of Richard Schwartz.^[7]

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

, 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

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

1. ^A. S. Švarc, A volume invariant of coverings {{icon ru}}, 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. ^¹Pierre de la Harpe, [https://books.google.com/books?id=cRT01C5ADroC&pg=PA87 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, [https://books.google.com/books?id=FmRMgAV8JwoC&pg=PA467 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, [https://books.google.com/books?id=Um4bCAAAQBAJ&pg=PA47&dq=%22geometric+action%22+milnor&hl=en&sa=X&ved=0ahUKEwiGs8W-1ufOAhWBOyYKHcW_B1sQ6AEIKzAD#v=onepage&q=%22geometric%20action%22%20milnor&f=false 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