A special class among the countably infinite relational structures is the class of homogeneous structures. These are the structures where every finite partial isomorphism extends to a total automorphism. A countable set, the ordered rationals, and the random graph are all homogeneous.
Given a homogeneous structure M, we will see some connections between its automorphism group (a Polish group) and some combinatorial properties of its age (the class of finite structures embedded in M). In particular, in a joint work with Solecki, we build on the work of Herwig-Lascar and Hodkinson-Otto by strengthening the notion of the extension property for partial automorphisms (EPPA) to the new notion of ‘coherent EPPA’. We will show that coherent EPPA implies the existence of a dense locally finite subgroup of Aut(M). We will also discuss other topological properties of Aut(M) such as ample generics and the small index property.

## Date:

Wed, 06/12/2017 - 11:00 to 13:00

## Location:

Math 209