Automorphisms of relatively hyperbolic groups and the Farrell–Jones conjecture

Abstract We prove the fibred Farrell–Jones conjecture (FJC) in A -, K -, and L -theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications: FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups; FJC is closed under extensions of FJC groups with kernel in a large class of relatively hyperbolic groups. Along the way we prove a number of results about JSJ decompositions of relatively hyperbolic groups which may be of independent interest.