Generalized John Gromov hyperbolic domains and extensions of maps

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Qingshan Zhou ◽  
Liulan Li ◽  
Antti Rasila
Keyword(s):  
John Domain ◽  
Gromov Hyperbolic ◽  
Uniformly Continuous ◽  
Gromov Boundary ◽  
Hyperbolic Domains

Let $\Omega \subset \mathbb{R}^n$ be a Gromov hyperbolic, $\varphi$-length John domain. We show that there is a uniformly continuous identification between the inner boundary of $\Omega$ and the Gromov boundary endowed with a visual metric, By using this result, we prove the boundary continuity not only for quasiconformal homeomorphisms, but also for more generally rough quasi-isometries between the domains equipped with the quasihyperbolic metrics.

scholarly journals Hyperbolic Unfoldings of Minimal Hypersurfaces

2018 ◽  
Vol 6 (1) ◽  
pp. 96-128 ◽  
Author(s):  
Joachim Lohkamp
Keyword(s):  
Point Of View ◽  
Hyperbolic Spaces ◽  
Intrinsic Geometry ◽  
Minimal Hypersurfaces ◽  
Bounded Geometry ◽  
New Concepts ◽  
Gromov Hyperbolic ◽  
Area Minimizing ◽  
Conformal Deformations ◽  
Gromov Boundary

Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

scholarly journals A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Hyperbolic Manifolds ◽  
Necessary Condition ◽  
Local Geometry ◽  
Gromov Hyperbolic ◽  
Relationship Of ◽  
The Relationship ◽  
Gromov Boundary

AbstractWe study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.

scholarly journals Random walks on weakly hyperbolic groups

2018 ◽  
Vol 2018 (742) ◽  
pp. 187-239 ◽  
Author(s):  
Joseph Maher ◽  
Giulio Tiozzo
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Linear Growth ◽  
Countable Group ◽  
Convergence Result ◽  
Hyperbolic Groups ◽  
Poisson Boundary ◽  
Gromov Hyperbolic ◽  
Gromov Boundary ◽  
Translation Length

Abstract Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

scholarly journals Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs

2018 ◽  
Vol 20 (05) ◽  
pp. 1750050 ◽  
Author(s):  
Alvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Martin Boundary ◽  
Hyperbolic Manifolds ◽  
Local Geometry ◽  
Gromov Hyperbolic ◽  
Relationship Of ◽  
The Relationship ◽  
Dirichlet Problem At Infinity ◽  
Gromov Boundary

In this paper, we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.

scholarly journals A density problem for Sobolev spaces on Gromov hyperbolic domains

2017 ◽  
Vol 154 ◽  
pp. 189-209 ◽  
Author(s):  
Pekka Koskela ◽  
Tapio Rajala ◽  
Yi Ru-Ya Zhang
Keyword(s):  
Sobolev Spaces ◽  
Gromov Hyperbolic ◽  
Density Problem ◽  
Hyperbolic Domains

scholarly journals The Boundary at Infinity of a Rough CAT(0) Space

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
S.M. Buckley ◽  
K. Falk
Keyword(s):  
Boundary Theory ◽  
Ideal Boundary ◽  
Common Features ◽  
Length Space ◽  
Gromov Hyperbolic ◽  
The Common ◽  
Gromov Boundary ◽  
The Ideal

Abstract We develop the boundary theory of rough CAT(0) spaces, a class of length spaces that contains both Gromov hyperbolic length spaces and CAT(0) spaces. The resulting theory generalizes the common features of the Gromov boundary of a Gromov hyperbolic length space and the ideal boundary of a complete CAT(0) space. It is not assumed that the spaces are geodesic or proper

scholarly journals Hyperbolic distance versus quasihyperbolic distance in plane domains

2021 ◽  
Vol 8 (20) ◽  
pp. 578-614
Author(s):  
David Herron ◽  
Jeff Lindquist
Keyword(s):  
Euclidean Plane ◽  
Metric Spaces ◽  
The Other ◽  
Hyperbolic Distance ◽  
Other Hand ◽  
Gromov Hyperbolic ◽  
Hyperbolic Domain ◽  
Hyperbolic Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

scholarly journals Cross ratios and cubulations of hyperbolic groups

2021 ◽  
Author(s):  
Jonas Beyrer ◽  
Elia Fioravanti
Keyword(s):  
Length Spectrum ◽  
Hyperbolic Groups ◽  
Optimal Length ◽  
Rigidity Result ◽  
Cube Complex ◽  
Geodesic Currents ◽  
Gromov Hyperbolic ◽  
Cube Complexes ◽  
The Relationship ◽  
Gromov Boundary

AbstractMany geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

2020 ◽  
Vol 4 (1) ◽  
pp. 29-39
Author(s):  
Dilrabo Eshkobilova ◽  
Keyword(s):  
Compact Support ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Continuous Maps ◽  
Uniformly Continuous

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

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