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We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic…
We prove that a non-elementary relatively hyperbolic group is statistically hyperbolic with respect to every finite generating set. We also establish statistical hyperbolicity for certain direct products of two groups, one of which is…
This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on…
We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
We study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a…
We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.
We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been…
A hyperbolic group acts by homeomorphisms on its Gromov boundary. We show that if this boundary is a topological n-sphere the action is topologically stable in the dynamical sense: any nearby action is semi-conjugate to the standard…
The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.
Bestvina-Feighn-Handel show that for finitely many generic and independent hyperbolic automorphisms $\phi_1, \cdots, \phi_r$ of $F_n$, the resulting extension $F_n \rtimes F_r$ is hyperbolic. This paper generalizes the above statement to…
Hyperbolicity is a property of a graph that may be viewed as being a "soft" version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic…
We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be…
In this note we revisit Moussong's Characterization of Gromov-hyperbolic Coxeter groups. A Coxeter group is Gromov-hyperbolic if and only if it does not contain a subgroup isomorphic to $\mathbb{Z}^2$ which can be read off directly from the…
Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the…
Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of…
This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a…
We classify all special homogeneous curves. A special homogeneous curve $\mathcal{H}$ consists of connected components of the hyperbolic points in the level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of degree at…
This article is devoted to the study of the historic set of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms…
We study the number of ends of a Schreier graph of a hyperbolic group. Let G be a hyperbolic group and let H be a subgroup of G. In general, there is no algorithm to compute the number of ends of a Schreier graph of the pair (G, H).…
The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph…