We study the countable set of rates of growth of a hyperbolic group
with respect to all its finite generating sets. We prove that the set is
well-ordered, and that every real number can be the rate of growth of at most
finitely many generating sets up to automorphism of the group.

We prove that the ordinal of the set of rates of growth is at least
$\omega^\omega$, and
in case the group is a limit group (e.g., free and surface groups), it is
$\omega^\omega$. This suggests a polynomial invariant of finite generating sets
of these limit groups.

We further study the rates of growth of all the finitely generated subgroups of
a hyperbolic group with respect to all their finite generating sets. This set is
proved to be well-ordered as well, and every real number can be the rate of
growth of at most finitely many isomorphism classes of finite generating sets of
subgroups of a given hyperbolic group. Finally, we strengthen our results to
include rates of growth of all the finite generating sets of all the
subsemigroups of a hyperbolic group.

Joint work with Koji Fujiwara.