Free subgroups and compact elements of connected Lie groups
M. I. Kabenyuk
Abstract:
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.
UDC:
512.5
MSC: Primary
22E20; Secondary
17B10,
22B05,
22C05,
22E25,
22E46 Received: 09.07.1983 and 19.10.1984