Generic properties of topological groups

Autor:	Elekes, Márton, Gehér, Boglárka, Kátay, Tamás, Keleti, Tamás, Kocsis, Anett, Pálfy, Máté
Rok vydání:	2022
Předmět:	Mathematics - Logic Mathematics - General Topology Mathematics - Group Theory 03E15 (Primary) 22B05 22C05 (Secondary)
Druh dokumentu:	Working Paper
Popis:	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. Comment: arXiv admin note: text overlap with arXiv:2110.15902
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2210.03034 Zobrazit plný text záznamu View this record from Arxiv