Vol. 10, No. 6, 2016

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
A probabilistic Tits alternative and probabilistic identities

Michael Larsen and Aner Shalev

Vol. 10 (2016), No. 6, 1359–1371
Abstract

We introduce the notion of a probabilistic identity of a residually finite group Γ. By this we mean a nontrivial word w such that the probabilities that w = 1 in the finite quotients of Γ are bounded away from zero.

We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable.

A main application of this result is a probabilistic variant of the Tits alternative: Let Γ be a finitely generated linear group over any field and let G be its profinite completion. Then either Γ is virtually solvable, or, for any n 1, n random elements g1,,gn of G freely generate a free (abstract) subgroup of G with probability 1.

We also prove other related results and discuss open problems and applications.

Keywords
Tits alternative, residually finite, virtually solvable, probabilistic identity, profinite completion
Mathematical Subject Classification 2010
Primary: 20G15
Secondary: 20E18
Milestones
Received: 29 October 2015
Revised: 1 May 2016
Accepted: 31 May 2016
Published: 30 August 2016
Authors
Michael Larsen
Department of Mathematics
Indiana University
Rawles Hall
Bloomington, IN 47405-5701
United States
Aner Shalev
Einstein Institute of Mathematics
The Hebrew University of Jerusalem
91904 Jerusalem
Israel