Journal of Lie Theory

no. 1
On the Component Factor Group G/G0 of a Pro-Lie Group G
Rafael Dahmen1; Karl H. Hofmann2
1Karlsruher Institut für Technologie, Hermann-von-Helmholtz-Platz 1, 76344 Karlsruhe-Eggenstein, Germany
2Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
Journal of Lie Theory, Volume 29 (2019) no. 1, pp. 221-225
A pro-Lie group G is a topological group such that G is isomorphic to the projective limit of all quotient groups G/N (modulo closed normal subgroups N) such that G/N is a finite dimensional real Lie group. A topological group is almost connected if the totally disconnected factor group Gt = G/G0 of G modulo the identity component G0 is compact. In this case it is straightforward that each Lie group quotient G/N of G has finitely many components. However, in spite of a comprehensive literature on pro-Lie groups, the following theorem, proved here, was not available until now:
Theorem. A pro-Lie group G is almost connected if each of its Lie group quotients G/N has finitely many connected components.
The difficulty of the proof is the verification of the completeness of Gt.
DOI: 10.5802/jolt.1054
Classification: 22A05, 22E15, 22E65, 22E99
Keywords: Pro-Lie groups, almost connected groups, projective limits

Rafael Dahmen  1 ; Karl H. Hofmann  2

1 Karlsruher Institut für Technologie, Hermann-von-Helmholtz-Platz 1, 76344 Karlsruhe-Eggenstein, Germany
2 Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany 
Rafael Dahmen; Karl H. Hofmann. On the Component Factor Group G/G0 of a Pro-Lie Group G. Journal of Lie Theory, Volume 29 (2019) no. 1, pp. 221-225. doi: 10.5802/jolt.1054
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