Automorphisms of graphs of groups with cyclic edge groups

TL;DR

This paper characterizes the outer automorphism group of certain one-ended graph of groups with cyclic edge groups, revealing its structure as virtually composed of automorphisms of vertex groups, associated Baumslag-Solitar groups, and specific twists.

Contribution

It provides a detailed description of the outer automorphism group for a class of graphs of groups with cyclic edge groups, extending understanding of their automorphism structures.

Findings

Outer automorphism group is virtually generated by vertex group automorphisms.

Includes automorphisms of associated generalized Baumslag-Solitar groups.

Introduces generalized twists as partial conjugations in the automorphism structure.

Abstract

We describe the outer automorphism group of a one-ended fundamental group of a graph of groups, when edge groups are cyclic, and vertex groups are torsion-free with cyclic centralizers. We show that in this case the outer automorphism group is virtually built from the outer automorphisms of the vertex groups (fixing some elements), the outer automorphisms of some associated generalized Baumslag-Solitar groups, and generalized twists - partial conjugations by elements in the centralizers of some elliptic elements in the group.