scholarly journals A universal Lipschitz extension property of Gromov hyperbolic spaces

2007 ◽  
pp. 861-896
Author(s):  
Alexander Brudnyi ◽  
Yuri Brudnyi
Keyword(s):  
Extension Property ◽  
Hyperbolic Spaces ◽  
Lipschitz Extension ◽  
Gromov Hyperbolic

scholarly journals Sobolev Extensions of Lipschitz Mappings into Metric Spaces

2017 ◽  
Vol 2019 (8) ◽  
pp. 2241-2265
Author(s):  
Scott Zimmerman
Keyword(s):  
Heisenberg Group ◽  
Bounded Domain ◽  
Compact Subset ◽  
Metric Space ◽  
Metric Spaces ◽  
Extension Property ◽  
Lipschitz Mapping ◽  
Lipschitz Extension ◽  
Lipschitz Mappings ◽  
Lipschitz Map

Abstract Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ can be extended to a $CL$-Lipschitz mapping on $\mathbb{R}^m$. In this article, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension $m$. We prove that any Lipschitz mapping from a compact subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ may be extended to a Sobolev mapping on any bounded domain containing the set. More generally, we prove this result in the case of mappings into any Lipschitz $(n-1)$-connected metric space.

scholarly journals FREELY INDECOMPOSABLE GROUPS ACTING ON HYPERBOLIC SPACES

2004 ◽  
Vol 14 (02) ◽  
pp. 115-171 ◽  
Author(s):  
ILYA KAPOVICH ◽  
RICHARD WEIDMANN
Keyword(s):  
Hyperbolic Group ◽  
Conjugacy Classes ◽  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Torsion Free

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups.

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 Rough isometry between Gromov hyperbolic spaces and uniformization

2021 ◽  
Vol 46 (1) ◽  
pp. 449-464
Author(s):  
Jeff Lindquist ◽  
Nageswari Shanmugalingam
Keyword(s):  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Rough Isometry

INFINITE WORDS AND CONFLUENT REWRITING SYSTEMS: ENDOMORPHISM EXTENSIONS

2009 ◽  
Vol 19 (04) ◽  
pp. 443-490 ◽  
Author(s):  
JULIEN CASSAIGNE ◽  
PEDRO V. SILVA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Spaces ◽  
Topological Structures ◽  
Infinite Words ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Gromov Hyperbolic ◽  
Characterization Theorems

Infinite words over a finite special confluent rewriting system R are considered and endowed with natural algebraic and topological structures. Their geometric significance is explored in the context of Gromov hyperbolic spaces. Given an endomorphism φ of the monoid generated by R, existence and uniqueness of several types of extensions of φ to infinite words (endomorphism extensions, weak endomorphism extensions, continuous extensions) are discussed. Characterization theorems and positive decidability results are proved for most cases.

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