  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 $\delta$-hyperbolic marked
groups whose entropy is bounded above by a number $H$ is finite with
cardinality depending on $\delta$ and $H$ alone. From these results, we draw
homotopical and topological finiteness theorems for compact metric spaces and
manifolds.
