Related papers: Gromov hyperbolic John is quasihyperbolic John I

In this note we prove that the symmetry of the quasi-isometry relation implies the axiom of choice, even when the relation is restricted to geodesic hyperbolic spaces. We show that this result is sharp by demonstrating that symmetry of…

The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.

We show that the homotopy groups of a Moore space $P^n(p^r)$, where $p^r \neq 2$, are $\mathbb{Z}/p^s$-hyperbolic for $s \leq r$. Combined with work of Huang-Wu, Neisendorfer, and Theriault, this completely resolves the question of when…

We characterize the class of Gromov hyperbolic spaces, whose boundary at infinity allow canonical M\"obius structures.

An improved version of quasiinvariance property of the quasihyperbolic metric under M\"obius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given. Next, a quasiinvariance property, sharp in a local sense, of the…

Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We will prove that under a small additional geometric assumption their notion of hyperbolicity is preserved by…

Classifying groups up to quasi-isometry is a fundamental problem in geometric group theory. In the context of hyperbolic and relatively hyperbolic groups, one of the key invariants in this classification is the boundary at infinity. F.…

We propose the study of Markov chains on groups as a "quasi-isometry invariant" theory that encompasses random walks. In particular, we focus on certain classes of groups acting on hyperbolic spaces including (non-elementary) hyperbolic and…

Behrstock, Hagen, and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups…

We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for…

Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still…

We compare a Gromov hyperbolic metric with the hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between the Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain…

We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.

For a compact negatively curved space, we develop a thermodynamic formalism framework to study the space of quasimorphisms of its fundamental group modulo bounded functions. We prove that this space is Banach isomorphic to the space of…

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric…

Here we give an extended review of the quasilinear reformulation of the Lovelock gravitational field equations in harmonic gauge presented in 1409.6656 [gr-qc]. This is important in order to establish rigorously well-posedness of the theory…

We show that in an L-annularly linearly connected, N-doubling, complete metric space, any n points lie on a K-quasi-circle, where K depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split…

Uniformization theory of Gromov hypebolic spaces investigated by Bonk, Heinonen and Koskela, generalizes the case where a classical Poincar\'e ball type model is used as the starting point. In this paper, we develop this approach in the…

Petyt-Spriano-Zalloum recently developed the notion of a \textit{curtain model}, which is a hyperbolic space associated to any CAT(0) space. It plays a similar role for CAT(0) spaces that curve graphs do for mapping class groups of…

We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and…