Graphs of subgroups of free groups

TL;DR

This paper introduces an efficient graph model for finitely generated subgroups of free groups, providing a concise proof of a key conjecture and addressing a question on subgroup intersections.

Contribution

It presents a novel, efficient graph-based model for subgroups and offers new proofs and answers related to the Hanna Neumann Conjecture and subgroup intersection ranks.

Findings

Short proof of the Amalgamated Graph Conjecture

Answer to Culler and Shalen's question on subgroup ranks

Independent confirmation by R. P. Kent IV

Abstract

We construct an efficient model for graphs of finitely generated subgroups of free groups. Using this we give a very short proof of Dicks's reformulation of the strengthened Hanna Neumann Conjecture as the Amalgamated Graph Conjecture. In addition, we answer a question of Culler and Shalen on ranks of intersections in free groups. The latter has also been done independently by R. P. Kent IV.