Stretching factors, metrics and train tracks for free products
Stretching factors, metrics and train tracks for free products
In this paper, we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.
We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.
We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).
Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.
A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.
859-899
Francaviglia, Stefano
72ccac56-d5b5-4689-b689-ef59b1f0647d
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Francaviglia, Stefano
72ccac56-d5b5-4689-b689-ef59b1f0647d
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Francaviglia, Stefano and Martino, Armando
(2017)
Stretching factors, metrics and train tracks for free products.
Illinois Journal of Mathematics, 59 (4), .
(doi:10.1215/ijm/1488186013).
Abstract
In this paper, we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.
We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.
We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).
Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.
A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.
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Submitted date: 11 November 2015
Accepted/In Press date: 25 August 2016
e-pub ahead of print date: 27 February 2017
Organisations:
Pure Mathematics
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Local EPrints ID: 406445
URI: http://eprints.soton.ac.uk/id/eprint/406445
ISSN: 0019-2082
PURE UUID: ee6c595c-daf2-49f3-8c5d-a0a87e9fb22b
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Date deposited: 10 Mar 2017 10:47
Last modified: 16 Mar 2024 03:59
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Author:
Stefano Francaviglia
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