Train track maps on graphs of groups

TL;DR

This paper develops a theory of train track maps on graphs of groups, extending existing concepts and proving that under certain conditions, homotopy equivalences can be represented by relative train track maps.

Contribution

It introduces a new framework for train track maps on graphs of groups and proves representation results under specific hypotheses.

Findings

Homotopy equivalences can be represented by relative train track maps under certain conditions.

The theory applies to graphs of groups with finite edge groups and certain generalized Baumslag-Solitar groups.

Provides a foundation for further study of automorphisms of groups via train track maps.

Abstract

In this paper we develop the theory of train track maps on graphs of groups. Expanding a definition of Bass, we define a notion of a map of a graph of groups, and of a homotopy equivalence. We prove that under one of two technical hypotheses, any homotopy equivalence of a graph of groups may be represented by a relative train track map. The first applies in particular to graphs of groups with finite edge groups, while the second applies in particular to certain generalized Baumslag-Solitar groups.