Descriptive group theory

In Chapter 1, we introduce a notion of universality for subgroups of Polish groups that has both algebraic and topological aspects. More precisely, given a class C of subgroups of a topological group Γ, we say that a subgroup H ∈ C is a universal C subgroup of Γ if every subgroup K ∈ C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural pre-order on the set of subgroups of Γ. In Chapter 2, we show that for any Polish group Γ, the countable power Γ has a universal analytic subgroup. Moreover, if Γ is locally compact, then Γ also contains universal Kσ and compactly generated subgroups. We prove a weaker version of this in the non-locally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universal analytic, Kσ and compactly generated subgroups. As an aside, we explore the relationship between the classes of Kσ and compactly generated subgroups and give conditions under which the two coincide. In Chapter 3, we study universal dense and co-null sets for the classes of Gδ and Fσ sets, respectively. Specifically, one says that A ⊂ X × Y is a universal dense Gδ for Y (resp. co-null Fσ) if the vertical cross-sections Ax are exactly the dense Gδ (resp. conull Fσ) subsets of Y . We discuss the relatioship between selection theorems for the product space X × Y and the existence of such universal sets. In the process, we refine a selection theorem of Debs and Saint-Raymond [2]. These results relate to a question of R. D. Mauldin [11].