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Deformation spaces and irreducible automorphisms of a free product

Deformation spaces and irreducible automorphisms of a free product
Deformation spaces and irreducible automorphisms of a free product
The (outer) automorphism group of a finitely generated free group Fn, which we denote by Out(Fn), is a central object in the fields of geometric and combinatorial group theory. My thesis focuses on the study of the automorphism group of a free product of groups. As every finitely generated group can be written as a free product of finitely many freely indecomposable groups and a finitely generated free group (Grushko’s Theorem) it seems interesting to study the outer automorphism group of groups that split as a free product of simpler groups. Moreover, it turns out that many well known methods for the free case, can be used for the study of the outer automorphism group of such a free product. Recently, Out(Fn) is mainly studied via its action on a contractible space (which is called Culler - Vogtmann space or outer space and we denote it by CVn)and a natural asymmetric metric which is called the Lipschitz metric. More generally, similar objects exist for a general non-trivial free product. In particular, in this thesis we generalise theorems that are well known for CVn and Out(Fn) in the case of a finite free product, using the appropriate definitions and tools.

Firstly, in [30], we generalise for an automorphism of a free product, a theorem due to Bestvina, Feighn and Handel, which states that the centraliser in Out(Fn) of an irreducible with irreducible powers automorphism of a free group is virtually infinite cyclic, where it is well known irreducible automorphisms form a (generic) class of automorphisms in the free case.
In [31], we use the previous result in order to prove that the stabiliser of an attractive fixed point of an irreducible with irreducible powers automorphism in the relative boundary of a free product, can be computed. This was already well known for the free case and it is a result of Hilion.
Finally, in [29] we prove that the Lipschitz metric for the general outer space is not even quasi-symmetric, but there is a ’nice’ function that bounds the asymmetry. As an application, we can see that this metric is quasi-symmetric if it is restricted on the thick part of outer space. The result in the free case is due to Algom-Kfir and Bestvina.
Syrigos, Dionysios
e698e7fe-fb8a-44e8-a9b5-972f772260c1
Syrigos, Dionysios
e698e7fe-fb8a-44e8-a9b5-972f772260c1
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1

Syrigos, Dionysios (2016) Deformation spaces and irreducible automorphisms of a free product. University of Southampton, School of Mathematics, Doctoral Thesis, 128pp.

Record type: Thesis (Doctoral)

Abstract

The (outer) automorphism group of a finitely generated free group Fn, which we denote by Out(Fn), is a central object in the fields of geometric and combinatorial group theory. My thesis focuses on the study of the automorphism group of a free product of groups. As every finitely generated group can be written as a free product of finitely many freely indecomposable groups and a finitely generated free group (Grushko’s Theorem) it seems interesting to study the outer automorphism group of groups that split as a free product of simpler groups. Moreover, it turns out that many well known methods for the free case, can be used for the study of the outer automorphism group of such a free product. Recently, Out(Fn) is mainly studied via its action on a contractible space (which is called Culler - Vogtmann space or outer space and we denote it by CVn)and a natural asymmetric metric which is called the Lipschitz metric. More generally, similar objects exist for a general non-trivial free product. In particular, in this thesis we generalise theorems that are well known for CVn and Out(Fn) in the case of a finite free product, using the appropriate definitions and tools.

Firstly, in [30], we generalise for an automorphism of a free product, a theorem due to Bestvina, Feighn and Handel, which states that the centraliser in Out(Fn) of an irreducible with irreducible powers automorphism of a free group is virtually infinite cyclic, where it is well known irreducible automorphisms form a (generic) class of automorphisms in the free case.
In [31], we use the previous result in order to prove that the stabiliser of an attractive fixed point of an irreducible with irreducible powers automorphism in the relative boundary of a free product, can be computed. This was already well known for the free case and it is a result of Hilion.
Finally, in [29] we prove that the Lipschitz metric for the general outer space is not even quasi-symmetric, but there is a ’nice’ function that bounds the asymmetry. As an application, we can see that this metric is quasi-symmetric if it is restricted on the thick part of outer space. The result in the free case is due to Algom-Kfir and Bestvina.

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Published date: July 2016
Organisations: University of Southampton, Mathematical Sciences

Identifiers

Local EPrints ID: 401557
URI: http://eprints.soton.ac.uk/id/eprint/401557
PURE UUID: 5f6c93ac-a742-4846-baeb-531e838a8101
ORCID for Dionysios Syrigos: ORCID iD orcid.org/0000-0002-7876-2641
ORCID for Armando Martino: ORCID iD orcid.org/0000-0002-5350-3029

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Date deposited: 27 Oct 2016 14:34
Last modified: 27 Jul 2019 00:21

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Contributors

Author: Dionysios Syrigos ORCID iD
Thesis advisor: Armando Martino ORCID iD

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