#### 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
##### Abstract

We introduce the notion of a probabilistic identity of a residually finite group $\Gamma$. By this we mean a nontrivial word $w$ such that the probabilities that $w=1$ in the finite quotients of $\Gamma$ 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 $\Gamma$ be a finitely generated linear group over any field and let $G$ be its profinite completion. Then either $\Gamma$ is virtually solvable, or, for any $n\ge 1$, $n$ 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
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