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 Ent(G, S) < E, we have |S| ≤ 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 δ-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. Contents 24 2.4. Non-geometric 3-manifolds 27 2.5. Ramified coverings 28 2.6. Higher dimensional graphs and cusp decomposable manifolds 29 Appendix A. Acylindrical splittings of hyperbolic 2-orbifolds 30 Appendix B. 3-manifolds with prescribed fundamental group 31 Appendix C. Malnormal subgroups of CAT(0)-groups 34 References 35

Ping-pong in Hadamard manifolds

“…Let X be an n-dimensional negatively curved Hadamard manifold, with sectional curvature ranging between −κ 2 and 1, for some κ ≥ 1. The main result of this note is the following quantitative version of the Tits alternative for X, which answers a question asked by Filippo Cerocchi during the Oberwolfach Workshop "Differentialgeometrie im Grossen", 2017, see also [10]: Theorem 1.1. There exists a function L = L(n, κ) such that the following holds: Let f, g be non-elliptic isometries of X generating a nonelementary discrete subgroup Γ of Isom(X).…”