Glasnik Matematicki, Vol. 55, No. 2 (2020), 301-336.

THE MINIMALLY DISPLACED SET OF AN IRREDUCIBLE AUTOMORPHISM IS LOCALLY FINITE

Stefano Francaviglia, Armando Martino and Dionysios Syrigos

Dipartimento di Matematica, University of Bologna, Italy
e-mail: stefano.francaviglia@unibo.it

Mathematical Sciences, University of Southampton, United Kingdom
e-mail: A.Martino@soton.ac.uk

Mathematical Sciences, University of Southampton, United Kingdom
e-mail: D.Syrigos@soton.ac.uk

Abstract.   We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

2010 Mathematics Subject Classification.   20E06, 20E36, 20E08

Key words and phrases.   Relative outer space, min set, automorphims of free products of groups, irreducible automorphisms

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.2.09

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