We present a theorem on the existence of local continuous
homomorphic inverses of surjective Borel homomorphisms with
countable kernels from Borel groups onto Polish groups. We also
associate in a canonical way subgroups of $\mathbb{R}$ with
certain analytic P-ideals of subsets of $\mathbb{N}$ . These
groups, with appropriate topologies, provide examples of Polish,
nonlocally compact, totally disconnected groups for which global
continuous homomorphic inverses exist in the situation described
above. The method of producing these groups generalizes
constructions of Stevens and Hjorth and, just as those
constructions, yields examples of Polish groups which are totally
disconnected and yet are generated by each neighborhood of the
identity.