A probabilistic Tits alternative and probabilistic identities

Larsen, Michael; Shalev, Aner

doi:10.2140/ant.2016.10.1359

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A probabilistic Tits alternative and probabilistic identities

Michael Larsen and Aner Shalev

Vol. 10 (2016), No. 6, 1359–1371

DOI: 10.2140/ant.2016.10.1359

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 $n \geq 1$ , $n$ random elements $g_{1}, \dots, g_{n}$ 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

Vol. 10, No. 6, 2016