Intersections of subgroups in virtually free groups and virtually free products

TL;DR

This paper generalizes the Friedman--Mineyev theorem to virtually free groups, providing bounds on the rank of intersections of free subgroups and extending the results to virtually free products.

Contribution

It offers a short proof of a generalized Hanna Neumann-type inequality for virtually free groups and extends the result to virtually free products.

Findings

Proved a generalized Hanna Neumann inequality for virtually free groups.

Extended the intersection bounds to virtually free products.

Provided a concise proof of the generalized theorem.

Abstract

This note contains a (short) proof of the following generalisation of the Friedman--Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $rank(A\cap B)-1\leqslant n\cdot(rank(A)-1)\cdot(rank(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.