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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Free products of topological groups with central amalgamation. II


Authors: M. S. Khan and Sidney A. Morris
Journal: Trans. Amer. Math. Soc. 273 (1982), 417-432
MSC: Primary 22A05; Secondary 20E06, 54D50
DOI: https://doi.org/10.1090/S0002-9947-1982-0667154-7
MathSciNet review: 667154
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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$.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0667154-7
Article copyright: © Copyright 1982 American Mathematical Society