On the Component Factor Group G/G₀ of a Pro-Lie Group G

2018/12/12 by Dahmen, Rafael, Hofmann, Karl-Heinrich
#22A05 #22E15 #22E65 #22E99 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1812.04838

Abstract

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 G_t:= 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: 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 G_t.

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