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
Contents
This result goes back, in different form, before the notion of quasi-isometry was formally introduced, to the work of Albert S. Schwarz (1955) and John Milnor (1968). Pierre de la Harpe called the Švarc–Milnor lemma ``the fundamental observation in geometric group theory" 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.
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).
Let
Then the group
is a quasi-isometry.
Here
Explanation of the terms
Recall that a metric
An action of
is finite.
The action of
Examples of applications of the Švarc–Milnor lemma
For Examples 1 through 5 below see pp. 89–90 in the book of de la Harpe. Example 6 is the starting point of the part of the paper of Richard Schwartz.
1. For every
2. If
3. If
4. If
5. If
6. If If