# Švarc-Milnor 引理

## 完整論述

${\displaystyle f_{p}:(G,d_{S})\to X,\quad g\mapsto gp}$

## 例子

1. 對任何${\displaystyle n\geq 1}$${\displaystyle \mathbb {Z} ^{n}}$呢個羣擬等距同構於歐幾里得空間${\displaystyle \mathbb {R} ^{n}}$
2. 如果${\displaystyle \Sigma }$係一個負歐拉特徵數嘅閉連通定向曲面嘅話，佢嘅基礎羣${\displaystyle \pi _{1}(\Sigma )}$同雙曲空間${\displaystyle \mathbb {H} ^{2}}$擬等距同構。
3. 如果${\displaystyle (M,g)}$係一個閉連通光滑流形，佢嘅基礎羣${\displaystyle \pi _{1}(M)}$${\displaystyle ({\tilde {M}},d_{\tilde {g}})}$擬等距同構，呢度${\displaystyle {\tilde {M}}}$${\displaystyle M}$萬有覆疊空間${\displaystyle {\tilde {g}}}$${\displaystyle g}$拉返上去${\displaystyle {\tilde {M}}}$${\displaystyle d_{\tilde {g}}}$就係由黎量度量${\displaystyle {\tilde {g}}}$誘導嘅路徑度量。
4. 如果${\displaystyle G}$係一個連通有限維李羣，帶著一個左不變黎曼度量同對應路徑度量，而${\displaystyle \Gamma \leq G}$係一個一致晶格嘅話，${\displaystyle \Gamma }$${\displaystyle G}$就係擬等距同構。
5. 如果${\displaystyle M}$係一個閉雙曲3-流形，咁${\displaystyle \pi _{1}(M)}$就同${\displaystyle \mathbb {H} ^{3}}$擬等距同構。
6. 如果${\displaystyle M}$係一個完備有限體積有尖嘅雙曲3-流形，咁${\displaystyle \Gamma =\pi _{1}(M)}$就同${\displaystyle \Omega =\mathbb {H} ^{3}-{\mathcal {B}}}$擬等距同構，呢度${\displaystyle {\mathcal {B}}}$係一柞${\displaystyle \Gamma }$-不變嘅極限球面（horoball），${\displaystyle \Omega }$就帶著誘導嘅路徑度量。

## 參考

1. A. S. Švarc, A volume invariant of coverings （俄文）, 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