Measures induced by sugroups and tuples in free groups

TL;DR

This paper investigates probability measures induced by homomorphisms from subgroups of free groups to finite groups, establishing conditions for profinite rigidity of subgroups based on these measures and extending previous results.

Contribution

It introduces a generalized framework for profinite rigidity using measures induced by tuples of generators, extending the concept from single words to arbitrary tuples.

Findings

Finitely generated subgroups are profinitely rigid iff their generating tuples are.

Generalizes previous results on word measures to tuples of generators.

Provides a new characterization of subgroup rigidity via induced measures.

Abstract

We study probability measure on $\mathrm{Hom}(H,G)$, where $G$ is a finite group and $H$ a finitely generated subgroup of a finitely generated free group $F$, obtained by pushing forward the uniform random homomorphisms $\mathrm{Hom}(F,G)$ via restriction map to $\mathrm{Hom}(H,G)$. This framework generalizes the word measures arising from single elements of a free group. We formalize the notion of profinite rigidity for subgroups via these induced measures. Our main result shows that a finitely generated subgroup is profinitely rigid if and only if any (equivalently, every) ordered generating tuple is profinitely rigid, thereby extending the notion of rigidity from individual word maps to arbitrary tuples. We also obtain a generalization of a result of \cite{puder2015measure}.