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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks
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by Stefano Francaviglia and Armando Martino PDF
Trans. Amer. Math. Soc. 374 (2021), 3215-3264 Request permission


This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification – the free splitting complex – with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes.

In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb {R}$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call partial train tracks (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement – minpoints – coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map.

In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

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Additional Information
  • Stefano Francaviglia
  • Affiliation: Dipartimento di Matematica of the University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
  • MR Author ID: 677681
  • Email:
  • Armando Martino
  • Affiliation: Mathematical Sciences, University of Southampton Highfield, Southampton SO17 1BJ, United Kingdom
  • MR Author ID: 646503
  • Email:
  • Received by editor(s): June 27, 2017
  • Received by editor(s) in revised form: May 17, 2018, February 11, 2020, and June 17, 2020
  • Published electronically: February 24, 2021
  • Additional Notes: Note: the two papers were originally packed together in the preprint arXiv:1703.09945. We decided to split that paper following the recommendations of a referee.
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 3215-3264
  • MSC (2020): Primary 20E06, 20E36, 20E08, 20F65
  • DOI:
  • MathSciNet review: 4237947