Alternating and Symmetric Separability of Free Products

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.