Title:

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 nontrivial 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 quasisymmetric, but there is a ’nice’ function that bounds the asymmetry. As an application, we can see that this metric is quasisymmetric if it is restricted on the thick part of outer space. The result in the free case is due to AlgomKfir and Bestvina.
