On a conjecture of Karrass and Solitar

TL;DR

This paper proves a longstanding conjecture by Karrass and Solitar, establishing a criterion for finite index subgroups in free products based on their intersection with normal subgroups, extending to subgroups of finite Kurosh rank.

Contribution

It provides a complete proof of the conjecture, linking finite index subgroups with their intersections with normal subgroups in free products and generalizing to finite Kurosh rank.

Findings

Finitely generated subgroups of free products have finite index iff they intersect all non-trivial normal subgroups.

The result applies more broadly to subgroups of finite Kurosh rank.

The conjecture of Karrass and Solitar is fully settled.

Abstract

We settle an old conjecture of Karrass and Solitar by proving that a finitely generated subgroup of a non-trivial free product $G = A\ast B$ has finite index if and only if it intersects non-trivially each non-trivial normal subgroup of $G$. This holds, more generally, for subgroups of finite Kurosh rank.