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 $ω^ω$,
and in case the group is a limit group (e.g., free and surface groups), it
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
Joint work with Koji Fujiwara.
- Topology Seminar