Alternating and Symmetric Separability of Free Products

TL;DR

This paper establishes conditions under which subgroups of free products of free and LERF groups are separable by alternating or symmetric groups, extending previous results to broader classes of groups.

Contribution

It provides new sufficient conditions for subgroup separability in free products involving free and LERF groups, generalizing Wilton's result.

Findings

Subgroups of free products can be separated using alternating or symmetric groups.

Finitely generated infinite-index subgroups of free groups are separable in certain free products.

The results apply to a broad class of free products involving LERF groups.

Abstract

Let $F \ast G$ be a free product of a free group $F$ and a LERF group $G$. In this note, we provide sufficient conditions for a subgroup $H$ of $F \ast G$ to be $\mathcal{A} \cup \mathcal{S}$-separable, that is, for any finite set $\{\gamma_1, \ldots, \gamma_n\} \subset (F \ast G) \setminus H$, there is a surjection $f$ from $F \ast G$ to an alternating or symmetric group such that $f(\gamma_i) \notin f(H)$ for all $i$. As a corollary, any finitely generated infinite-index subgroup of a free group is $\mathcal{A} \cup \mathcal{S}$-separable in the free product of the free group and an arbitrary LERF group, generalizing a result of Wilton.