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.

Bibliographie

[1] Djokovič D. Ž., “The union of compact subgroups of a connected locally compact group”, Math. Z., 158 (1978), 99–105 | DOI | MR | Zbl

[2] Kabenyuk M. I., “Svyaznye gruppy s plotnymi mnozhestvami kompaktnykh elementov”, Ukr. mat. zhurn., 33 (1981), 179–183 | MR | Zbl

[3] Platonov V. P., “K kategorii topologicheskikh grupp”, DAN SSSR, 162:4 (1965), 755–758 | MR | Zbl

[4] Wang S. P., “On density properties of certain subgroups with boundedness conditions”, Monatsh. Math., 89 (1980), 141–162 | DOI | MR | Zbl

[5] Tits Zh., “Svobodnye podgruppy lineinykh grupp”, Matematika (sb. perevodov), 16 (1972), 47–66 | Zbl

[6] Dauns J., A concrete approach to division rings, Heldermann Verlag, Berlin, 1982 | MR | Zbl

[7] Adams Dzh., Lektsii po gruppam Li, Nauka, M., 1979 | MR

[8] Goto M., Grosskhans F., Poluprostye algebry Li, Mir, M., 1981 | MR | Zbl

[9] Veil G., Klassicheskie gruppy, ikh invarianty i predstavleniya, IL, M., 1947

[10] Ragunatan M., Diskretnye podgruppy grupp Li, Mir, M., 1977 | MR

[11] Malcev A. I., “On the theory of the Lie groups in the large”, Matem. sb., 16(58):2 (1945), 163–190 | MR | Zbl

[12] Kreknin V. A., Kostrikin A. I., “Ob algebrakh Li s regulyarnym avtomorfizmom”, DAN SSSR, 149:2 (1963), 249–251 | MR | Zbl

[13] Kreknin V. A., “O razreshimosti algebr Li s regulyarnym avtomorfizmom konechnogo poryadka”, DAN SSSR, 150:3 (1963), 467–469 | MR | Zbl

[14] Burbaki N., Gruppy i algebry Li, gl. VII, VIII, Mir, M., 1978 | MR

[15] Borel A., “On free subgroups of semi-simple groups”, L'Enseig. Math., 29 (1983), 151–164 | MR | Zbl

[16] Kabenyuk M. I., “Kompaktnye elementy i svobodnye podgruppy svyaznykh grupp Li”, XVII Vsesoyuzn. algebr, konf., Chast 1, In-t matem. AN BSSR, Minsk, 1983, 85