The idea of self-similarity is one of the most fundamental in the modern mathematics. The
notion of “renormalization group”, which plays an essential role in quantum field theory …

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an ℝ–
tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic …

This paper presents a study of the asymptotic geometry of groups with contracting elements,
with emphasis on a subclass of statistically convex-cocompact (SCC) actions. The class of …

Let delta _\mathcal S, λ denote the Floyd metric on a discrete group G generated by a finite
set\mathcal S with respect to the scaling function fn= λ n for a positive λ< 1. We prove that if …

For a random walk on a finitely generated group G we obtain a generalization of a classical
inequality of Ancona. We deduce as a corollary that the identity map on G extends to a …

We describe the kernel of the canonical map from the Floyd boundary of a relatively
hyperbolic group to its Bowditch boundary. Using the Floyd completion we further prove that …

We study random walks on metric spaces with contracting isometries. In this first article of the
series, we establish sharp deviation inequalities by adapting Gou\" ezel's pivotal time …

We provide characterizations of Anosov representations of word hyperbolic groups into real
semisimple Lie groups in terms of the existence of equivariant limit maps on the Gromov …

Let a discrete group G act by homeomorphisms of a compactum in a way that the action is
properly discontinuous on triples and cocompact on pairs. We prove that such an action is …

We study different notions of quasiconvexity for a subgroup H of a relatively hyperbolic
group G. Our first result implies that relative geometric quasiconvexity is equivalent to …