Latent symmetry of graphs and stretch factors in Out$(F_r)$

Abstract

Every irreducible outer automorphism of the free group of rank $r$ is topologically represented by an irreducible train track map $f:\Gamma \to \Gamma$ for 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.