Amalgamated free products of topological groups being Hausdorff -- a new approach

TL;DR

This paper investigates the conditions under which the free product with amalgamation of Hausdorff topological groups remains Hausdorff, providing a new characterization and conditions for the topology to be Hausdorff.

Contribution

It introduces a new approach to characterize the topology of amalgamated free products and establishes conditions for them to be Hausdorff, especially when the amalgamated subgroup is open.

Findings

The topology coincides with the $X_0$-topology in Mal'tsev's sense.

Canonical mappings are open homeomorphic embeddings when the subgroup is open.

The amalgamated free product is Hausdorff if the amalgamated subgroup is open.

Abstract

The paper deals with the problem posed by Katz and Morris whether the free product with amalgamation of any Hausdorff topological groups is Hausdorff, the negative solution of which (even for the particular case of a closed amalgamated subgroup) easily follows from the relevant result by Uspenskij. The topology of such a product is characterized by proving that it coincides with the so-called $X_0$-topology in the sense of Mal'tsev for the corresponding pushout $X$ in the category of Hausdorff topological spaces. Applying this characterization, it is proved that the canonical mappings of Hausdorff groups into their amalgamated free product are open homeomorphic embeddings if an amalgamated subgroup is open. This immediately implies that in that case this product is Hausdorff.