Free products of topological groups with amalgamation
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.
scientific article; zbMATH DE number 3929314