Free subgroups and compact elements of connected Lie groups

M. I. Kabenyuk

Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 273-283

Résumé

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