We prove that there exists a positive, explicit function such that, for any group admitting a -acylindrical splitting and any generating set of with , we have . We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, -quasiconvex-malnormal amalgamated products acting on -hyperbolic spaces or on -spaces with entropy bounded by . 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 -manifolds, higher dimensional graph manifolds and cusp-decomposable manifolds, ramified coverings and, more generally, -groups with negatively curved splittings.
Cite this article
Filippo Cerocchi, Andrea Sambusetti, Entropy and finiteness of groups with acylindrical splittings. Groups Geom. Dyn. 15 (2021), no. 3, pp. 755–799DOI 10.4171/GGD/611