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

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 fact...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Author: Hillen, Paige
Format: Article
Language:English
Subjects:
Automorphisms
Graph theory
Graphical representations
Isomorphism
Lower bounds
Symmetry
Vertex sets
Online Access:Get full text
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Summary: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.
ISSN:2331-8422