Latent symmetry of graphs and stretch factors in Out$(F_{r})$

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