Taking the High-Edge Route Through Outer Space in Rank 3

TL;DR

This paper investigates the properties of principal fully irreducible outer automorphisms of rank-3 graphs, identifying which graphs support train track maps and how these automorphisms relate to certain geometric structures in Outer space.

Contribution

It characterizes the rank-3 graphs that admit train track maps for principal fully irreducible outer automorphisms and describes the associated Outer space simplices they pass through.

Findings

Identifies rank-3 graphs supporting train track maps for principal automorphisms.

Determines which Outer space simplices are traversed by principal axes.

Establishes connections between automorphism properties and geometric structures.

Abstract

Principal outer automorphisms were introduced in Algom-Kfir-Kapovich-Pfaff to emulate principal pseudo-Anosov surface homeomorphisms, i.e. those whose attracting and repelling invariant foliations have only 3-pronged singularities. It is proved in Algom-Kfir-Kapovich-Pfaff that principal fully irreducible outer automorphisms semi-mimic principal pseudo-Anosov surface homeomorphisms in important ways. We prove here which rank-3 graphs carry train track maps for principal fully irreducible outer automorphisms. As a corollary, one obtains which rank-3 Culler-Vogtmann Outer space simplices are passed through by principal axes.