The Hanna Neumann Conjecture and the rank of the join

TL;DR

This paper investigates the Hanna Neumann conjecture's extension to infinite index subgroups, providing counterexamples and demonstrating the effectiveness of a graph-based method for generating subgroups with non-trivial intersections.

Contribution

It extends the Hanna Neumann conjecture to the finite index case, shows its failure in the general case, and applies a graph-based method to generate relevant counterexamples.

Findings

The conjecture holds only for finite index subgroups.

Counterexamples exist for the general case.

Graph-based methods effectively generate subgroups with complex intersections.

Abstract

The Hanna Neumann conjecture gives a bound on the intersection of finitely generated subgroups of free groups. We explore a natural extension of this result, which turns out to be true only in the finite index case, and provide counterexamples for the general case. We also see that the graph-based method of generating random subgroups of free groups developed by Bassino, Nicaud and Weil is well-suited to generating subgroups with non-trivial intersections. The same method is then used to generate a counterexample to a similar conjecture of Guzman.