Universal Subgroups of Polish Groups

Given a class C of subgroups of a topological group G, we say that a subgroup H ∈ C is a universal C subgroup of G if every subgroup K ∈ C is a continuous homomorphic pre-image 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 G. We show that for any locally compact Polish group G, the countable power G has a universal Kσ subgroup and a universal compactly generated subgroup. 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 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. 2010 Mathematics Subject Classification. 03E15, 54H11, 22A05, 20B30, 20B35. E-mail address. [email protected]