Product set growth in groups and hyperbolic geometry

We discuss product theorems in groups acting on hyperbolic spaces:
for every hyperbolic group there exists a constant a > 0 such that for
every finite subset U that is not contained in a virtually cyclic subgroup,
| U^3 | > ( a | U | )^2 . We also discuss the growth of | U^n | and
conclude that the entropy of U (the limit of 1/n log | U^n | as n goes
to infinity) exceeds 1/2 log ( a | U | ) . This generalizes results of
Razborov and Safin, and answers a question of Button. We discuss similar
estimates for groups acting
acylindrically on trees or hyperbolic spaces. This talk is on a joint
work with T. Delzant.