Free products of topological groups with amalgamation
type: Item
instance of: scholarly article
title: Free products of topological groups with amalgamation
zbMATH Open document ID: 0581.22002
author: Elyahu Katz
author: Sidney A. Morris
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: When topological groups \(G_ 1\) and \(G_ 2\) have a common subgroup A, the free product of \(G_ 1\) and \(G_ 2\) with amalgamated subgroup A is the topological group \(G_ 1 *_ A G_ 2\) which has these properties: \(G_ 1\) and \(G_ 2\) are subgroups and their union generates the group algebraically; and every pair of continuous morphisms \(\phi_ i: G_ i\to D\) into a topological group D which agree on A can be extended to a continuous morphism of \(G_ 1 *_ A G_ 2\) into D. It is proved here that if each \(G_ i\) is a \(k_{\omega}\)-group and A is compact, then \(G_ 1 *_ A G_ 2\) is a \(k_{\omega}\)-group, and in particular is Hausdorff. The subgroup A needs not to be normal, as has been required in earlier theorems of this sort by these and other authors. The present result covers a large class of examples, since every locally compact connected Hausdorff topological group is a \(k_{\omega}\)-group.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 22D05
zbMATH DE Number: 3929314
zbMATH Keywords: topological groups
zbMATH Keywords: free product
zbMATH Keywords: amalgamated subgroup
zbMATH Keywords: \(k_{\omega }\)- group
MaRDI profile type: Publication
full work available at URL: pjm.1985.119.169
OpenAlex ID: W2063460724
DOI: 10.2140/PJM.1985.119.169
description: scientific article; zbMATH DE number 3929314