Entropy and finiteness of groups with acylindrical splittings

Abstract

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 $\mathrm{Ent}(G,S)$, we have $|S|\le F(k,E)$. 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 $\delta$-hyperbolic spaces or on $\mathrm{CAT}(0)$-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, $\mathrm{CAT}(0)$-groups with negatively curved splittings.