University of Sydney Algebra Seminar

Colin Reid (University of Newcastle)

Friday 25 Sep, 12-1pm, Place: Carslaw 375

Normal subgroup structure in Polish groups

I will present joint work with Phillip Wesolek (UCLouvain, Belgium). The isomorphism theorems of group theory have deficiencies in the setting of topological groups, because an injective continuous group homomorphism is not in general a closed map. This is an obstacle to developing a well-behaved theory of series of closed normal subgroups in topological groups. We develop some theory to overcome these complications in the context of Polish groups, that is, topological groups in which the topology is separable and completely metrizable. The key observation in the Polish context is that given an injective, continuous homomorphism \(\psi: G \rightarrow H\) whose image is dense and normal, then the corresponding action of \(H\) on \(G\) via conjugation of \(\psi(G)\) is jointly continuous, meaning that the corresponding semidirect product \(G \rtimes H\) is a topological group with the product topology. Closely related to this is the notion of a quasi-product (a generalization of direct products previously studied by Caprace--Monod) and an equivalence relation on non-abelian chief factors that allows us to give a uniqueness result for chief factors in finite normal series for Polish groups.