Characterizing Lie groups by controlling their zero-dimensional subgroups

Abstract

AbstractWe provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows:(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.(ii) An abelian topological groupGis a Lie group if and only ifGis locally minimal, locally precompact and all closed metric zero-dimensional subgroups ofGare discrete.(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.