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

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.