Švarc–Milnor lemma: Difference between revisions
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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 <math>G</math>, equipped with a "nice" [[discrete action|discrete]] isometric [[ |
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 <math>G</math>, equipped with a "nice" [[discrete action|discrete]] isometric [[Group action (mathematics)|action]] on a [[metric space]] <math>X</math>, is [[Quasi-isometry|quasi-isometric]] to <math>X</math>. |
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⚫ | This result goes back, in different form, before the notion of [[quasi-isometry]] was formally introduced, to the work of [[Albert Schwarz|Albert S. Schwarz]] (1955)<ref>A. S. Švarc, ''A volume invariant of coverings'' {{ |
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⚫ | This result goes back, in different form, before the notion of [[quasi-isometry]] was formally introduced, to the work of [[Albert Schwarz|Albert S. Schwarz]] (1955)<ref>A. S. Švarc, ''A volume invariant of coverings'' {{in lang|ru}}, [[Doklady Akademii Nauk SSSR]], vol. 105, 1955, pp. 32–34.</ref> and [[John Milnor]] (1968).<ref>J. Milnor, ''A note on curvature and fundamental group'', [[Journal of Differential Geometry]], vol. 2, 1968, pp. 1–7</ref> Pierre de la Harpe called the Švarc–Milnor lemma "the ''fundamental observation in geometric group theory''"<ref name=D>Pierre 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</ref> 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 [[Benson Farb|Farb]] and Margalit.<ref>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</ref> |
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==Precise statement== |
==Precise statement== |
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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 |
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).<ref name=BH>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}}</ref> |
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Then the group <math>G</math> is finitely generated and for every finite generating set <math>S</math> of <math>G</math> and every point <math>p\in X</math> |
Then the group <math>G</math> is finitely generated and for every finite generating set <math>S</math> of <math>G</math> and every point <math>p\in X</math> |
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Here <math> d_S</math> is the [[word metric]] on <math>G</math> corresponding to <math>S</math>. |
Here <math> d_S</math> is the [[word metric]] on <math>G</math> corresponding to <math>S</math>. |
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⚫ | Sometimes a properly discontinuous cocompact isometric action of a group <math>G</math> on a proper geodesic metric space <math>X</math> is called a ''geometric'' action.<ref>I. Kapovich, and N. Benakli, [https://books.google.com/books?id=Um4bCAAAQBAJ&dq=%22geometric+action%22+milnor&pg=PA47 ''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</ref> |
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==Notes== |
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In many sources the Švarc–Milnor lemma is stated under a slightly more restrictive assumption that the space <math>X</math> be a [[geodesic metric space]] (rather that a [[length space]]), and most applications concern this context. |
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⚫ | Sometimes a properly discontinuous cocompact isometric action of a group <math>G</math> on a proper geodesic metric space <math>X</math> is called a ''geometric'' action.<ref>I. Kapovich, and N. Benakli, [https://books.google.com/books?id=Um4bCAAAQBAJ |
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==Explanation of the terms== |
==Explanation of the terms== |
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Recall that a metric <math> X</math> |
Recall that a metric space <math> X</math> is ''proper'' if every closed ball in <math>X</math> is [[Compact space|compact]]. |
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An action of <math>G</math> on <math> X</math> is ''properly discontinuous'' if for every compact <math>K\subseteq X</math> the set |
An action of <math>G</math> on <math> X</math> is ''properly discontinuous'' if for every compact <math>K\subseteq X</math> the set |
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:<math>\{g\in G |
:<math>\{g\in G \mid gK\cap K\ne \varnothing\}</math> |
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is finite. |
is finite. |
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The action of <math>G</math> on <math> X</math> is ''cocompact'' if the quotient space <math>X/G</math>, equipped with the [[quotient topology]], is compact. |
The action of <math>G</math> on <math> X</math> is ''cocompact'' if the quotient space <math>X/G</math>, equipped with the [[quotient topology]], is compact. |
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Under the other assumptions of the Švarc–Milnor lemma, the cocompactness condition is equivalent to the existence of a closed ball <math>B</math> in <math>X</math> such that |
Under the other assumptions of the Švarc–Milnor lemma, the cocompactness condition is equivalent to the existence of a closed ball <math>B</math> in <math>X</math> such that |
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:<math>\ |
:<math>\bigcup_{g\in G} gB=X.</math> |
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==Examples of applications of the Švarc–Milnor lemma== |
==Examples of applications of the Švarc–Milnor lemma== |
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For Examples 1 through 5 below see pp. |
For Examples 1 through 5 below see pp. 89–90 in the book of de la Harpe.<ref name=D/> |
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Example 6 is the starting point of the part of the paper of [[Richard Schwartz]]<ref>[[Richard Schwartz]], ''The quasi-isometry classification of rank one lattices'', Publications Mathématiques de l'Institut des Hautes Études Scientifiques, vol. 82, 1995, pp. |
Example 6 is the starting point of the part of the paper of [[Richard Schwartz (mathematician)|Richard Schwartz]].<ref>[[Richard Schwartz (mathematician)|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</ref> |
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⚫ | # If <math>(M,g)</math> is a closed connected smooth manifold with a smooth [[Riemannian metric]] <math>g</math> then <math>\pi_1(M)</math> is quasi-isometric to <math>(\tilde M, d_{\tilde g})</math>, where <math>\tilde M</math> is the [[universal cover]] of <math>M</math>, where <math> \tilde g </math> is the pull-back of <math> g</math> to <math>\tilde M</math>, and where <math> d_{\tilde g}</math> is the path metric on <math>\tilde M</math> defined by the Riemannian metric <math> \tilde g </math>. |
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⚫ | # If <math>G</math> is a connected finite-dimensional [[Lie group]] equipped with a left-invariant [[Riemannian metric]] and the corresponding path metric, and if <math>\Gamma\le G</math> is a [[Lattice (discrete subgroup)#Definition|uniform lattice]] then <math>\Gamma</math> is quasi-isometric to <math>G</math>. |
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⚫ | # If <math>M</math> is a complete finite volume hyperbolic 3-manifold with cusps, then <math>\Gamma=\pi_1(M)</math> is quasi-isometric to <math>\Omega= \mathbb H^3-\mathcal B</math>, where <math>\mathcal B </math> is a certain <math>\Gamma</math>-invariant collection of [[horoball]]s, and where <math>\Omega</math> is equipped with the induced path metric. |
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==References== |
==References== |
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{{Reflist}} |
{{Reflist|30em}} |
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{{DEFAULTSORT:Svarc-Milnor lemma}} |
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[[Category:Geometric group theory |
[[Category:Geometric group theory]] |
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[[Category:Metric geometry]] |
Latest revision as of 21:12, 3 May 2024
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
[edit]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 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
[edit]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
[edit]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]
- For every the group is quasi-isometric to the Euclidean space .
- If is a closed connected oriented surface of negative Euler characteristic then the fundamental group is quasi-isometric to the hyperbolic plane .
- 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 .
- 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 .
- If is a closed hyperbolic 3-manifold, then is quasi-isometric to .
- 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
[edit]- ^ A. S. Švarc, A volume invariant of coverings (in Russian), Doklady Akademii Nauk SSSR, vol. 105, 1955, pp. 32–34.
- ^ J. Milnor, A note on curvature and fundamental group, Journal of Differential Geometry, vol. 2, 1968, pp. 1–7
- ^ 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
- ^ 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
- ^ 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
- ^ 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
- ^ 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