Algebra & Number Theory
- Algebra Number Theory
- Volume 10, Number 6 (2016), 1359-1371.
A probabilistic Tits alternative and probabilistic identities
We introduce the notion of a probabilistic identity of a residually finite group . By this we mean a nontrivial word such that the probabilities that in the finite quotients of 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 be a finitely generated linear group over any field and let be its profinite completion. Then either is virtually solvable, or, for any , random elements of freely generate a free (abstract) subgroup of with probability .
We also prove other related results and discuss open problems and applications.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 20E18: Limits, profinite groups
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