Probabilistically nilpotent groups

TL;DR

This paper characterizes finitely generated residually finite groups where certain commutator words are probabilistic identities, showing they are virtually nilpotent of a specific class, thus linking probabilistic properties to algebraic structure.

Contribution

It establishes a precise equivalence between probabilistic identities and virtual nilpotency of a given class in finitely generated residually finite groups.

Findings

Probabilistic identities characterize virtually nilpotent groups.

The word $[x_1, \, \ldots, \, x_k]$ is a probabilistic identity iff the group is virtually nilpotent of class less than $k$.

Results extend to related contexts and open problems are discussed.

Abstract

We show that, for a finitely generated residually finite group $\Gamma$, the word $[x_1, \ldots, x_k]$ is a probabilistic identity of $\Gamma$ if and only if $\Gamma$ is virtually nilpotent of class less than $k$. Related results, generalizations and problems are also discussed.