Measures induced by sugroups and tuples in free groups

TL;DR

This work extends the study of finite-quotient measures from individual words to finitely generated subgroups of free groups by introducing the measures $\mu_{H,G}$ induced on $Hom(H,G)$ via restriction from $Hom(F_n,G)$. It establishes a tight link between subgroup rigidity and tuple rigidity: a subgroup $H\le F_n$ is profinitely rigid iff some (equivalently every) ordered generating tuple of $H$ is profinitely rigid as a tuple. Using profinite completion techniques and a LERF framework, the authors unify and generalize prior results on word measures (including primitive words) and provide structural corollaries for subgroups generated by powers of generators. The results clarify how the ensemble of finite-quotient statistics determines subgroup structure up to automorphisms of the ambient free group, and they extend rigidity phenomena to broader classes of subgroups and free-product decompositions, with potential applications to distinguishing subgroups by their finite quotients.

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}.

Theorems & Definitions (22)

• Theorem 1
• Corollary 1
• Definition 2
• Theorem 2
• Definition 3
• Theorem 3
• proof
• Lemma 4: Zapirain, Proposition 3.2
• Proposition 5
• proof