Entropy and finiteness of groups with acylindrical splittings

We prove that there exists a positive, explicit function F.k; E/such that, for any group G admitting a k-acylindrical splitting and any generating set S of G with Ent.G; S/< E, we have jSj ≤ F.k; E/. We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, D-quasiconvex k-malnormal amalgamated products acting on ı-hyperbolic spaces or on CAT.0/-spaces with entropy bounded by E. A number of finiteness results for interesting families of Riemannian or metric spaces with bounded entropy and diameter also follow: Riemannian 2-orbifolds, non-geometric 3-manifolds, higher dimensional graph manifolds and cusp-decomposable manifolds, ramified coverings and, more generally, CAT.0/-groups with negatively curved splittings.