Group extensions and discrete subgroups of Lie groups

Abstract

Let F be a discrete uniform subgroup of a connected simply connected solvable Lie group S. It is shown how S is essentially determined by F, using the point of view of group extensions. Let r be a discrete uniform subgroup of a connected simply connected solvable Lie group S. The main purpose of this work is to study how S is essentially determined by T from the point of view of group extensions. In [5], Mal'cev proved that T determines 5 uniquely if S is nilpotent. However, for a general solvable group, the situation seems less favorable. Nevertheless, L. Auslander (Theorem 2 in [1]) has obtained some results when T is the fundamental group of a nilmanifold, and in the subsequent works of Auslander and Tolimieri ([2], [8], [9]), stronger results were proved by using the notation of semisimple splittings, from which one can obtain the generalization of the result in [1] to arbitrary discrete uniform subgroups T. It is the purpose of this paper to revisit and prove directly this generalization by making use of the theory of group extensions as Auslander originally did in [1]. The author is indebted to the referee for pointing out the above results of Auslander and Tolimieri. 1. Let G and H be topological groups. An extension of G by H is a pair (E, rr) consisting of a topological group E which contains G as a closed normal subgroup and a continuous open homomorphism it of E onto H whose kernel is G. Two extensions (Et, 7r,-) of G by H are said to be equivalent if there exists an isomorphism a : Ex-+E2 of topological groups which leaves elements of G fixed and is such that tt2o=ttx. If (E, it) is an extension of G by H, then this determines a homomorphism Tr°:H—>0(G), where 0(G) denotes the group A(G) of all automorphisms of G modulo the inner automorphism group 1(G) of G. If (Ex, irx) and (E2, tt2) are equivalent, then 77^=712. For any homomorphism -0(G), let Ext(G, H, cp) denote the set of all equivalence classes of the extensions (E, it) of G by H Received by the editors July 12, 1971 and, in revised form, November 17, 1971. AMS 1970 subject classifications. Primary 22E15, 22E20, 22E40.