CTs for free products

TL;DR

This paper develops a theory of CT train track maps for outer automorphisms of free products, extending concepts from free group automorphisms and establishing new properties like index inequalities and fixed subgroup results.

Contribution

It introduces a framework for representing automorphisms of free products with CT maps, extending train track theory and analyzing automorphism properties in this context.

Findings

Outer automorphisms satisfy an index inequality similar to known results.

Automorphisms can be represented by CT maps after taking a positive power.

Fixed subgroup properties analogous to Culler’s theorem are established.

Abstract

The fundamental group of a finite graph of groups with trivial edge groups is a free product. We are interested in those outer automorphisms of such a free product that permute the conjugacy classes of the vertex groups. We show that in particular cases of interest, such as where vertex groups are themselves finite free products of finite and cyclic groups, given such an outer automorphism, after passing to a positive power, the outer automorphism is represented by a particularly nice kind of relative train track map called a CT. CTs were first introduced by Feighn and Handel for outer automorphisms of free groups. We develop the theory of attracting laminations for and principal automorphisms of free products. We prove that outer automorphisms of free products satisfy an index inequality reminiscent of a result of Gaboriau, Jaeger, Levitt and Lustig and sharpening a result of Martino. Finally, we prove a result reminiscent of a theorem of Culler on the fixed subgroup of an automorphism of a free product whose outer class has finite order.