Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups

Gérard Besson (1) , G Courtois (2) , S Gallot (1) , A Sambusetti (3)

1 IF - Institut Fourier
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free δ-hyperbolic marked groups whose entropy is bounded above by a number H is finite with cardinality depending on δ and H alone. We then draw topological consequences of these results.

Domaines: Géométrie métrique [math.MG]; Géométrie différentielle [math.DG]; Théorie des groupes [math.GR]

Gérard Besson, G Courtois, S Gallot, A Sambusetti. Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups. 2021. ⟨hal-03355476v2⟩