Algebra & Number Theory

A probabilistic Tits alternative and probabilistic identities

Michael Larsen and Aner Shalev

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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

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

Received: 29 October 2015
Revised: 1 May 2016
Accepted: 31 May 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Tits alternative residually finite virtually solvable probabilistic identity profinite completion


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.

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