On probabilistic identities and coset identities in pro-$p$ groups

TL;DR

This paper demonstrates that probabilistic identities in certain non-archimedean analytic groups are actually coset identities, and explores their implications for pro-$p$ groups and related structures using Lie-theoretic methods.

Contribution

It establishes that probabilistic identities in specific groups are equivalent to coset identities and applies this to prove a probabilistic Tits alternative for certain pro-$p$ groups.

Findings

Probabilistic identities imply coset identities in non-archimedean analytic groups.

Certain pro-$p$ groups satisfy a probabilistic Tits alternative.

Lie-theoretic methods reveal properties of torsion probabilistic identities.

Abstract

It is shown that a probabilistic identity on a $\sigma$-compact $K$-analytic group $G$, $K$ a non-archimedean local field, is a coset identity. As an application, one concludes that compact groups which are linear over a local field and various pro-$p$ groups obtained from free constructions satisfy a probabilistic Tits alternative. By means of Lie-theoretic methods, we also study torsion probabilistic identities in virtually free pro-$p$ and compact $p$-adic analytic groups.