Algorithmic constructions of relative train track maps and CTs

TL;DR

This paper proves that the construction of CTs, a type of relative train track map representing rotationless outer automorphisms of free groups, can be performed algorithmically, enabling practical computations in this area.

Contribution

It introduces an algorithmic method for constructing CTs and checking reduced inclusions of free factor systems, advancing computational techniques in geometric group theory.

Findings

Algorithmic construction of CTs established.

Decidable criterion for reduced free factor system inclusion.

Applications to automorphism analysis and free group dynamics.

Abstract

Every rotationless outer automorphism of a finite rank free group is represented by a particularly useful relative train track map called a CT. The main result of this paper is that the constructions of CTs can be made algorithmic. A key step in our argument is proving that it is algorithmic to check if an inclusion of one invariant free factor system in another is reduced. Several applications are included, as well as algorithmic constructions for relative train track maps in the general case.