Tuesday, October 30, 2018, 10.00
Seminar room N0.003, Mathematical Institute, University of Bonn
Speaker: Aleksandra Kwiatkowska (Münster)
Title: Infinite permutation groups
We discuss several results on infinite permutation groups, that is, closed subgroups of the symmetric group on a countable set, or equivalently, automorphism groups of countable structures. We will focus on ample genericity, where a topological group G has ample generics if for every n, the diagonal conjugacy action of G on Gn has a comeager orbit, on similarity classes, and on topological generators of permutation groups. For example, we show that for a permutation group G, under mild assumptions, for every n and an n-tuple f in G, the countable group generated by f is discrete, or precompact, or the conjugacy class of f is meager. Finally, we will focus on automorphism groups of structures equipped with a definable linear order, such as the ordered random graph, the ordered rational Urysohn metric space, the ordered random poset, the ordered random boron tree, and many other extremely amenable permutation groups. In particular, we give new examples of such groups which have a comeager conjugacy class. This is joint work with Maciej Malicki.