Separability and Randomness in Free Groups

TL;DR

This paper establishes new subgroup separability results in free groups using probabilistic methods, showing that certain subgroups can be distinguished via homomorphisms onto alternating groups.

Contribution

It introduces probabilistic techniques to prove subgroup separability in free groups, extending previous deterministic approaches.

Findings

Existence of homomorphisms onto alternating groups distinguishing subgroups

Probabilistic counting of fixed points under these homomorphisms

New separability results for finitely generated subgroups of free groups

Abstract

We prove new separability results about free groups. Namely, if $H_1, \ldots , H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$, then there exists a homomorphism onto some alternating group $f:F \twoheadrightarrow A_m$ such that whenever $H_i$ is not conjugate into $H_j$, then $f(H_i)$ is not conjugate into $f(H_j)$. The proof is probabilistic. We count the expected number of fixed points of $f(H_i)$'s and their subgroups under a carefully constructed measure.