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
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] Several minor variations of the statement of the lemma exist in the literature.
Here we follow the version given in the book of Bridson and Haefliger (see Proposition 8.19 on p. 140 there).
be a group acting by isometries on a proper length space
Sometimes a properly discontinuous cocompact isometric action of a group
, 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
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.