Švarc-Milnor theorem

The Švarc-Milnor theorem (in other transcriptions also the Schwartz-Milnor theorem or Schwarz-Milnor theorem) is a mathematical theorem from the field of geometric group theory . It was named after the mathematicians Albert S. Švarc and John W. Milnor .

statement

Let be a geodetic metric space in which closed spheres with finite radius are compact. The topological group operates co- compactly on and for all compact sets the set is finite. ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle K\subset X}$ ${\displaystyle \{g\in G\mid gK\cap K\neq \emptyset \}}$

Then it is finitely generated and for each the mapping is a quasi-isometry with respect to the word metrics defined for an (arbitrary) finite generating system . ${\displaystyle G}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle G\to X,g\mapsto g.x_{0}}$

Examples

• The group of whole numbers is quasi-isometric to the real number line . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {R} }$
• The group is quasi-isometric to . ${\displaystyle \mathbb {Z} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$
• The fundamental group of a compact metric space is quasi-isometric to the universal superposition (if this exists). ${\displaystyle \pi _{1}X}$ ${\displaystyle X}$ ${\displaystyle {\widetilde {X}}}$

literature

• AS Švarc: A volume invariant of covering , Dokl. Akad. Nauka SSSR (NS), 105 (1955), 32-34.
• JW Milnor: A note on curvature and fundamental group , J. Differential Geometry, 2: 1-7 (1968). on-line

Web links

• The Svarc-Milnor Lemma