Latent symmetry of graphs and stretch factors in Out(Fr)

TL;DR

This paper establishes bounds on the stretch factors of irreducible outer automorphisms of free groups using the concept of latent symmetry, and classifies certain train track maps based on these symmetries.

Contribution

It introduces the notion of latent symmetry to analyze and classify irreducible train track maps with a single fold, providing new bounds on stretch factors.

Findings

Lower bounds on stretch factors in terms of folds and edges

Latent symmetry classifies all irreducible single fold train track maps

Precise bounds when the map is periodic on vertices

Abstract

Every irreducible outer automorphism of the free group of rank r is topologically represented by an irreducible train track map $f$ on some graph $\Gamma$ of rank r. Moreover, $f$ can always be written as a composition of folds and a graph isomorphism. We give a lower bound on the stretch factor of an irreducible outer automorphism in terms of the number of folds of $f$ and the number of edges in $\Gamma$. In the case that $f$ is periodic on the vertex set of $\Gamma$, we show a precise notion of the latent symmetry of $\Gamma$ gives a lower bound on the number of folds required. We use this notion of latent symmetry to classify all possible irreducible single fold train track maps.