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 \operatorname{Ent}(G,S) , 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 \operatorname{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, \operatorname{CAT}(0) -groups with negatively curved splittings.
Highlights
In this paper we are interested in finitely generated groups G admitting k-acylindrical splittings, that is isomorphic to the fundamental group of a graph of groups such that the action of G on the corresponding Bass–Serre tree is k-acylindrical
We recall that an action without inversions of a finitely generated group on a simplicial tree T is said to be k-acylindrical if the fixed point set of any element g 2 G has diameter at most k
We present some basic examples of application of Theorem 1 to particular classes of spaces whose groups naturally possess acylindrical splittings and presentations with an uniform bound on the acylindricity constant and on the length of relators
Summary
In this paper we are interested in finitely generated groups G admitting k-acylindrical splittings, that is isomorphic to the fundamental group of a graph of groups such that the action of G on the corresponding Bass–Serre tree is (non elementary and) k-acylindrical. Notice that, while the results on Riemannian 2-orbifolds and ramified coverings (though concerning metrics of any possible sign) still pertain to the framework of spaces of negative curvature, the classes of non-geometric 3-manifolds and of high dimensional graph or cusp decomposable manifolds escape from the realm of non-positive curvature. This is clear for non-prime 3-manifolds, and follows from Leeb’s work [40] for irreducible 3-manifolds.
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