An analogue of the strengthened Hanna Neumann conjecture for virtually free groups and virtually free products

TL;DR

This paper extends the Friedmann--Mineyev theorem, originally for free groups, to virtually free groups and groups with free product structures, providing new bounds on subgroup intersections.

Contribution

It introduces an analogue of the Hanna Neumann inequality for virtually free groups and groups with free product structures, broadening the scope of the original conjecture.

Findings

Established an inequality for subgroup intersections in virtually free groups.

Generalized the Friedmann--Mineyev theorem to broader classes of groups.

Provided theoretical bounds applicable to groups with free product structures.

Abstract

The Friedman--Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup).