Locally compact automorphism groups of first order structures

Dugald Macpherson, University of Leeds. Part of the Algebra, Logic, and Algorithms Seminar Series.

An automorphism group of a countable structure M is locally compact if there is a finite subset F of M such that the group of automorphisms of M fixing F pointwise  has all its orbits finite.  I will discuss several results and questions around such groups.  In joint work with Praeger and Smith, we use a Hrushovski amalgamation construction to show that for every k there is a (non-discrete) locally compact automorphism group which is k-transitive but not (k+1)-transitive.  We ask for an example of a non-discrete locally compact subgroup of the symmetric group which is maximal subject to being closed, and we give a related classification result.  I will also discuss maximal closed subgroups of the symmetric group on a countably infinite set and connections to categoricity.