Automorphisms of graphs of cyclic splittings of free groups

TL;DR

This paper proves that all symmetries of certain graphs representing cyclic splittings of free groups are derived from automorphisms of the groups themselves, extending to various types of splittings.

Contribution

It establishes that isometries of graphs of cyclic splittings are exactly induced by outer automorphisms of free groups, covering multiple splitting types.

Findings

Isometries of the graphs correspond to outer automorphisms.

The result applies to graphs of maximally-cyclic and very small splittings.

The theorem holds for free groups of rank at least 3.

Abstract

We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings.