## Algebra & Number Theory

### A probabilistic Tits alternative and probabilistic identities

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 6 (2016), 1359-1371.

Dates
Revised: 1 May 2016
Accepted: 31 May 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842555

Digital Object Identifier
doi:10.2140/ant.2016.10.1359

Mathematical Reviews number (MathSciNet)
MR3544299

Zentralblatt MATH identifier
1356.20030

Subjects
Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 20E18: Limits, profinite groups

#### Citation

Larsen, Michael; Shalev, Aner. A probabilistic Tits alternative and probabilistic identities. Algebra Number Theory 10 (2016), no. 6, 1359--1371. doi:10.2140/ant.2016.10.1359. https://projecteuclid.org/euclid.ant/1510842555

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