Relative ends, ℓ2-invariants and property (T)

We establish a splitting theorem for one-ended groups H⩽G such that and the almost malnormal closure of H is a proper subgroup of G. This yields splitting theorems for groups G with non-trivial first ℓ2-Betti number . We verify the Kropholler Conjecture for pairs H⩽G satisfying . We also prove that every n-dimensional Poincaré duality (PDn) group containing a PDn−1 group H with property (T) splits over a subgroup commensurable with H.