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Generic properties of topological groups

Generic properties of topological groups

Authors :
Elekes, Márton
Gehér, Boglárka
Kátay, Tamás
Keleti, Tamás
Kocsis, Anett
Pálfy, Máté
Publication Year :
2022

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.<br />Comment: arXiv admin note: text overlap with arXiv:2110.15902

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2210.03034
Document Type :
Working Paper