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 $k$-acylindrical splitting and generating set $S$ of with $\mathrm{Ent}(G,S)<E$, we have $|S| \leq F(k, E)$. deduce corresponding finiteness results classes groups possessing acylindrical splittings acting geometrically bounded entropy: instance, $D$-quasiconvex $k$-malnormal amalgamated products on $\delta$-hyperbolic spaces or $CAT(0)$-spaces entropy by $E$. A number interesting families Riemannian metric diameter also follow: 2-orbifolds, non-geometric $3$-manifolds, higher dimensional graph manifolds cusp-decomposable manifolds, ramified coverings and, more generally, CAT(0)-groups negatively curved splittings.