FREE SUBGROUPS AND COMPACT ELEMENTS OF CONNECTED LIE GROUPS

Abstract

Let be the set of compact (i.e., contained in some compact subgroup) elements of a topological group , and let be its closure. The following assertions are proved:Theorem 1. A compact connected semisimple Lie group has a free dense subgroup each of whose nonidentity elements is a generator of a maximal torus in .Theorem 2. Suppose that a connected Lie group has no nontrivial compact elements in its center and coincides with the closure of its commutator group, and let be its Lie algebra. The following conditions are equivalent: (i) . (ii) has a dense subgroup of compact elements. (iii) , where is a nilpotent ideal and is a semisimple compact algebra whose adjoint action on does not have a zero weight. (iv) , where is a nilpotent connected simply connected normal subgroup and is a semisimple compact connected subgroup whose center acts (by conjugations) regularly on .Corollary. A locally compact connected group that coincides with the closure of its commutator group has a dense subgroup of compact elements if and only if .Bibliography: 16 titles.