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A strong Schottky Lemma for nonpositively curved singular spaces

Summary: A strong Schottky Lemma for nonpositively curved
singular spaces
To John Stallings on his sixty-fifth birthday
Roger C. Alperin, Benson Farb
, and Guennadi A. Noskov
January 9, 2001
1 Introduction
The classical Schottky Lemma (due to Poincare, Klein, Schottky) gives a
criterion for a pair of isometries g, h of hyperbolic space to have powers
gm, hn which generate a free group. This criterion was generalized by Tits
to pairs of elements in linear groups in his proof of the Tits alternative.
In this paper we give a criterion (Theorem 1.1) for pairs of isometries of a
nonpositively curved metric space (in the sense of Alexandrov) to generate
a free group without having to take powers. This criterion holds only in
singular spaces, for example in Euclidean buildings; in fact our criterion
takes a particularly simple form in that case (Corollary 1.2).
The original motivation for our criterion was to prove that the four
dimensional Burau representation is faithful. While linearity of braid groups
is now known, this question is still open; it is well-known to be related to
detecting the unknot with the Jones polynomial . It was shown in [4] that the


Source: Alperin, Roger C. - Department of Mathematics, San Jose State University


Collections: Mathematics