Free products of topological groups with central amalgamation. II

Abstract

In Free products of topological groups with central amalgamation. I, we introduced the notion of amalgamated free product of topological groups and showed that if $A$ is a common central closed subgroup of Hausdorff topological groups $G$ and $H$, then the amalgamated free product $G{\coprod _A}H$ exists and is Hausdorff. In this paper, we give an alternative much shorter (but less informative) proof of this result. We then proceed to describe the properties of $G{\coprod _A}H$. In particular, we find necessary and sufficient conditions for $G{\coprod _A}H$ to be a locally compact Hausdorff group, a complete metric group, and a maximally almost periodic group. Properties such as being a Baire space and connectedness are also investigated. In the case that $G$, $H$ and $A$ are ${k_\omega }$-groups, the topology of $G{\coprod _A}H$ is fully described. A consequence of this description is that for ${k_\omega }$-groups $G{\coprod _A}H$ is homeomorphic to $(G{ \times _A}H) \times F(G/A\Lambda H/A)$, where $G{ \times _A}H$ is the direct product of $G$ and $H$ with $A$ amalgamated, and $F(G/A\Lambda H/A)$ is the free topological group on the smash product of $G/A$ and $H/A$.