Entropy and finiteness of groups with acylindrical splittings

TL;DR

This paper establishes explicit bounds on generating set sizes for groups with acylindrical splittings based on entropy, leading to finiteness results for various classes of groups and spaces with bounded entropy.

Contribution

It introduces an explicit function bounding generating set size for groups with acylindrical splittings and derives finiteness results for multiple geometric and topological group classes.

Findings

Bound on generating set size based on entropy and acylindrical splitting parameters

Finiteness results for groups acting on hyperbolic and CAT(0) spaces with bounded entropy

Applications to Riemannian orbifolds, 3-manifolds, and higher-dimensional manifolds

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 $\mathrm{Ent}(G,S)<E$, we have $|S| \leq F(k, E)$. 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 $CAT(0)$-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.