  We prove a quantitative version of the classical Tits' alternative for
discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex
geodesic bicombing. Some geometric consequences, as uniform estimates on
systole, diastole, algebraic entropy and critical exponent of the groups, will
be presented. Finally we will study the behaviour of these group actions under
limits, providing new examples of compact classes of metric spaces.
