Automorphism Groups of Homogeneous Structures

A homogeneous structure is a countable (finite or countably infinite) first order structure such that every isomorphism between finitely generated substructures extends to an automorphism of the whole structure. Examples of homogeneous structures include any countable set, the pentagon graph, the random graph, and the linear ordering of the rationals. Countably infinite homogeneous structures are precisely the Fraisse limits of amalgamation classes of finitely generated structures. Homogeneous structures and their automorphism groups constitute the main theme of the thesis.

The automorphism group of a countably infinite structure becomes a Polish group when endowed with the pointwise convergence topology. Thus, using Baire Category one can formulate the following notions. A Polish group has generic automorphisms if it contains a comeagre conjugacy class. A Polish group has ample generics if it has a comeagre diagonal conjugacy class in every dimension. To study automorphism groups of homogeneous structures as topological groups, we examine combinatorial properties of the corresponding amalgamation classes such as the extension property for partial automorphisms (EPPA), the amalgamation property with automorphisms (APA), and the weak amalgamation property. We also explain how these combinatorial properties yield the aforementioned topological properties in the context of homogeneous structures.

The main results of this thesis are the following. In Chapter 3 we show that any free amalgamation class over a finite relational language has Gaifman clique faithful coherent EPPA. Consequently, the automorphism group of the corresponding free homogeneous structure contains a dense locally finite subgroup, and admits ample generics and the small index property. In Chapter 4 we show that the universal bowtie-free countably infinite graph admits generic automorphisms. In Chapter 5 we prove that Philip Hall's universal locally finite group admits ample generics. In Chapter 6 we show that the universal homogeneous ordered graph does not have locally generic automorphisms. Moreover we prove that the universal homogeneous tournament has ample generics if and only if the class of finite tournaments has EPPA.