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 $H/A$.