Rates of growth in hyperbolic (and other) groups

In the late 1970s W. Thurston proved that the countable set of volumes of

closed hyperbolic 3-manifolds is well-ordered, that only finitely many closed hyperbolic

3-manifolds can have the same volume, and that the ordinal of the set of these volumes  is

${\omega_0}^{\omega_0}$.

We prove analogous results for the rates of growth of hyperbolic (and other) groups. 

We study the countable set of rates of growth of a hyperbolic group with respect to all

its finite sets of generators,  the countable set of rates of growth of all the finitely generated 

subgroups of a hyperbolic group (with respect to all their finite generating sets), and the rates

of growth of all the finitely generated subsemigroups of a hyperbolic group. 

Our results suggests a polynomial invariant for generating sets (and tuples) in (some) hyperbolic groups.

Joint work with Koji Fujiwara.