Unlike the two preceding papers [8,9] on this topic, we restrict attention here to Poincare duality groups over Z. Thus we use the term PD n -group to mean PD n -group over Z throughout the paper. We begin with a basic result. THEOREM A. Let G be a PD n -group, and suppose that G splits over a soluble-byfinite subgroup S. Then S is polycyclic-by-finite of Hirsch length n-\. Now every torsion-free polycyclic-by-finite group of Hirsch length m is a PD mgroup, and conversely, every soluble-by-finite PD n -group is torsion-free and polycyclic-by-finite. These facts are established in [2, Theorems 9.9, 9.10 and 9.23]. Thus if G is a PD n -group and 5 is a soluble-by-finite subgroup of G, then the question of whether or not G splits over S only arises if S is actually a PD n~1 -group. In this context, the authors [8,9] introduced a theory of obstructions to splittings, and so once Theorem A is established the questions about splittings are reduced to questions about the vanishing of these obstructions.The precise definition of the obstructions is given in [9]. In essence one has a group G which satisfies some special kind of duality, the archetypal example being the fundamental group of a compact aspherical manifold, and two PD n~1 -subgroups S and T. In this situation there is an obstruction sing (S, T) which depends heavily on the way S and T sit inside G. Our main theorems here apply to the special case when both S and T are polycyclic-by-finite. To state these results we must introduce the crucial concept of subgroups controlling singularities. In the following definition we write Comm G (r) = {g€G: Tand T 9 are commensurable}
Abstract. We show that a group G contains a subgroup K with e(G, K) > 1 if and only if it admits an action on a connected cube that is transitive on the hyperplanes and has no fixed point. As a corollary we deduce that a countable group G with such a subgroup does not satisfy Kazhdan's property (T).
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