7 research outputs found
Conformal dimension and random groups
We give a lower and an upper bound for the conformal dimension of the
boundaries of certain small cancellation groups. We apply these bounds to the
few relator and density models for random groups. This gives generic bounds of
the following form, where is the relator length, going to infinity.
(a) 1 + 1/C < \Cdim(\bdry G) < C l / \log(l), for the few relator model,
and
(b) 1 + l / (C\log(l)) < \Cdim(\bdry G) < C l, for the density model, at
densities .
In particular, for the density model at densities , as the relator
length goes to infinity, the random groups will pass through infinitely
many different quasi-isometry classes.Comment: 32 pages, 4 figures. v2: Final version. Main result improved to
density < 1/16. Many minor improvements. To appear in GAF
Growth gap in hyperbolic groups and amenability
International audienceWe prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane