Pro-Lie groups

Abstract

A topological group G is pro-Lie if G has small compact normal subgroups K such that G/K is a Lie group. A locally compact group G is an L-group if, for every neighborhood U of the identity and compact set C, there is a neighborhood V of the identity such that gHg1 n C C U for every g C G and every subgroup H C V. We obtain characterizations of pro-Lie groups and make several applications. For example, every compactly generated L-group is pro-Lie and a compactly generated group which can be embedded (by a continuous isomorphism) in a pro-Lie group is pro-Lie. We obtain related results for factor groups, nilpotent groups, maximal compact normal subgroups, and generalize a theorem of Hofmann, Liukkonen, and Mislove [4]. Introduction. We divide the paper into two sections. In ?1 we consider some rather general properties of locally compact groups and establish their relationships to pro-Lie groups. In ?2 we obtain some related results and consider pro-Lie groups in more special settings. We obtain several characterizations of pro-Lie groups, the last one in terms of the bounded part and periodic part of the group. In this paper we are concerned only with those groups which are locally compact. The following background is taken directly from the referee's report. The application of Lie group theory is the basis of most of the finer results on the structure theory of locally compact groups. Historically, the first half of this century saw topological group theory concentrate on the interaction of Lie group theory and the general theory of compact and locally compact groups. The guiding principle was one of Hilbert's problems formulated at the turn of the century (the famed number FIVE). Even before this program was completed, Iwasawa published a key paper in 1949 on the structure theory of locally compact groups which are projective limits of Lie groups, i.e. those groups which now are called somewhat unfortunately from the view point of good English-pro-Lie groups. The vitality of Iwasawa's ideas continues to influence writers in the area up to these days, as is, e.g., exemplified by this paper. The fine-structure theory of locally compact groups flourished through the sixties. While this work did not show the raw power of penetration of the work of Gleason, Montgomery, and Yamabe, which in the early fifties led to the solution of Hilbert's Fifth Problem, it contributed much to the insight into the structure of more special classes of locally compact groups through the contributions of Grosser, Hofmann, Moskowitz, Mostert and their followers and students. In the meantime one observes a steady, although somewhat slower, flow of results on locally compact groups, and their structure theory springs from the work of researchers in the field, while more Received by the editors November 23, 1983 and, in revised form, April 9, 1984. 1980 Mathematics S*ect Clwsification. Primary 22D05. (?)1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page