Hausdorff dimension of boundaries of relatively hyperbolic groups

This is a joint work with Wenyuan Yang (University of Bejing, China). In the study
of different types of boundaries of groups, the notion of the Hausdorff dimension
plays an important role. I would be happy to talk about this notion in the city where
F. Hausdorff has been living and working for many years.

The aim is to extend several classical results known in the theory of discrete groups
of hyperbolic spaces due to S. Patterson and D . Sullivan to relatively hyperbolic
groups. The latter ones admit convergence and geometrically finite actions by
homeomorphisms on compacta. This class of groups properly contains Gromov
hyperbolic groups (which admit geometrically finite actions on compacta but without
parabolic elements). M. Coornaert has generalized the results of Patterson-Sullivan
to the case of hyperbolic groups having proved that the critical exponent of their
Poincaré series coincides with the Hausdorff dimension of their Gromov boundary
(or limit set).

In the case of relatively hyperbolic groups their boundaries are in general not
homeomorphic to their limit sets (contrary to the hyperbolic groups). We consider
the Hausdorff dimension of them with respect to a Floyd metric which plays a crucial
role for these groups.

We prove that the classical result of Patterson-Sullivan-Coornaert is true for finitely
generated relatively hyperbolic groups: the Hausdorff dimension of the boundary/limit
set is equal to the growth rate of the group times a constant. Moreover an interesting
feature is that this dimension is achieved on a very "thin" subset of uniformly conical
points whose Hausdorff dimension is equal to that of the whole boundary/limit set.