On Central Topological Groups

Abstract

1. Introduction.Let G be a locally compact topological group and Z(G) its center.We shall be concerned with the class [Z] of all locally compact groups G such that G¡Z(G) is compact.It turns out that the various structural results on groups in [Z] generalize and unify in a natural manner those of compact groups on the one hand, and of locally compact abelian groups on the other.The present paper gives complete proofs of the results announced in Bull.Amer.Math.Soc.72 (1966), 831-836 under the same title.§2 contains various elementary conditions for groups G to be in [Z] and [SIN] and is preparatory in character.Theorems 3.1 and 3.2 of §3 are extension theorems for certain continuous homomorphisms <f>: N-+ V, where N is a closed normal subgroup of G, G e [Z], and Kis a vector group.An essential fact which we utilize is contained in Lemma 1, due to P. Cartier ([1, p. 22-01] or [5, p. 36]).Various conditions are imposed upon AT (or Z(G)) and upon 4>; the extension </c G->■ V of <j> will be either a crossed homomorphism with respect to a given representation p of G on F or a homomorphism (if no representation is involved).We show by an example that, in contrast to abelian group theory, torus-valued homomorphisms do not extend.An application of one of these extension theorems gives a direct estimate of the rank of the fundamental group of an analytic group in [Z].By sharpening the estimate given by P. Smith in [18, pp.210-229], we answer a question raised there.In order to facilitate the exposition we introduce the following terminology.We denote by 91 (or 91(G)) the group of topological group automorphisms of the locally compact group G. (As is well known, in the compact-open topology, 91 is not-in general-a topological group; a finer topology must be employed.)The group » of inner automorphisms of G is a normal (but not necessarily closed) subgroup of 91.If 95 is any subgroup of 91 we say that G has small 93-invariant neighborhoods of the identity if every neighborhood of the identity in G contains a 93-invariant neighborhood of the identity.In particular, if 95 = », we simply say that G has small invariant neighborhoods of the identity.The class of all such groups will be denoted by [SLV].Groups of this sort were first studied by G. Mostow in [13] and by K. Iwasawa in [7](3).