Relative ends, l2-invariants and property (T)

We establish a splitting theorem for one-ended groups H <= G such that (e) over tilde (G, H) >= 2 and the almost malnormal closure of H is a proper subgroup of C. This yields splitting theorems for groups G with non-trivial first l(2)-Betti number beta(2)(1)(G). We verify the Kropholler Conjecture for pairs H <= G satisfying beta(2)(1)(G) > beta(2)(1)(H) We also prove that every n-dimensional Poincare duality (PDn) group containing a PDn-1 group H with property (T) splits over a subgroup commensurable with H. (C) 2011 Elsevier Inc. All rights reserved.