Generic properties of topological groups

TL;DR

This paper explores the typical properties of various classes of topological groups, revealing that certain classes have a comeager isomorphism class while others do not, highlighting the diversity of generic behaviors.

Contribution

It establishes the existence of a comeager isomorphism class among countably infinite abelian and compact metrizable abelian groups, contrasting with the meagerness in other classes.

Findings

Comeager isomorphism class among countably infinite abelian groups.

Comeager isomorphism class among compact metrizable abelian groups.

Generic compact metrizable groups are neither connected nor totally disconnected.

Abstract

We study generic properties of topological groups in the sense of Baire category. First we investigate countably infinite (discrete) groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. I. Goldbring, S. E. Kunnawalkam and Y. Lodha proved that every isomorphism class is meager among countably infinite (discrete) groups. In contrast, we show that there is a comeager isomorphism class among countably infinite (discrete) abelian groups. Then we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite (discrete) case. Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.