UNIVERSAL SUBGROUPS OF POLISH GROUPS

Given a class C of subgroups of a topological group G, we say that a subgroup H is an element of C is a universal C subgroup of G if every subgroup K is an element of C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural preorder on the set of subgroups of G. We show that for any locally compact Polish group G, the countable power G(omega) has a universal K-sigma subgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally 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 K-sigma and compactly generated subgroups. As an aside, we explore the relationship between the classes of K-sigma and compactly generated subgroups and give conditions under which the two coincide.