General Burnside problem for locally compact groups

Abstract

A locally compact group G is called compactly covered (respectively compactly ruled) if it is an union of compact subgroups (resp. if it is a directed union of compact subgroups). The class $\left[\mathbf{CR}\right]$ of all compactly ruled groups is a proper subclass of the class $\left[\mathbf{CC}\right]$ of all compactly covered groups. In this paper we shall study the following question: For which classes $\mathfrak{X}$ of locally compact groups do the following holds: $\mathfrak{X}\cap \left[\mathbf{CC}\right]\subseteq \left[\mathbf{CR}\right].$

Keywords:

• Burnside problem
• compact exponent
• compactly covered group
• compactly ruled group
• locally compact group
• solvable group
• totally disconnected group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 22D05

Acknowledgments

We would like to thank the anonymous referee for the careful reading of the paper and for his/her valuable comments.