Vol. 10, No. 6, 2016

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A probabilistic Tits alternative and probabilistic identities

Michael Larsen and Aner Shalev

Vol. 10 (2016), No. 6, 1359–1371

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.

Tits alternative, residually finite, virtually solvable, probabilistic identity, profinite completion
Mathematical Subject Classification 2010
Primary: 20G15
Secondary: 20E18
Received: 29 October 2015
Revised: 1 May 2016
Accepted: 31 May 2016
Published: 30 August 2016
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