Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture

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: (1) FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups; (2) 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.