Harmonic quasi-isometric maps into Gromov hyperbolic $\operatorname{CAT}(0)$-spaces

2021 ◽  
Vol 118 (3) ◽  
Author(s):  
Hubert Sidler ◽  
Stefan Wenger
Keyword(s):  
Gromov Hyperbolic

Mathematical Properties of Gromov Hyperbolic Graphs

2010 ◽  
Author(s):  
Sergio Bermudo ◽  
José M. Rodríguez ◽  
José M. Sigarreta ◽  
Jean-Marie Vilaire ◽  
Theodore E. Simos ◽  
...  
Keyword(s):  
Mathematical Properties ◽  
Gromov Hyperbolic ◽  
Hyperbolic Graphs

scholarly journals Negative curvature in graph braid groups

2020 ◽  
pp. 1-36
Author(s):  
Anthony Genevois
Keyword(s):  
Braid Group ◽  
Negative Curvature ◽  
Braid Groups ◽  
Gromov Hyperbolic ◽  
Relatively Hyperbolic ◽  
Geometric Study

In this paper, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous paper, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic and acylindrically hyperbolic.

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 On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

2009 ◽  
Vol 61 (4) ◽  
pp. 740-761 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Frédéric Haglund
Keyword(s):  
Abelian Group ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Free Abelian Group ◽  
Gromov Hyperbolic ◽  
Convex Cocompact ◽  
Rank 2 ◽  
Cocompact Subgroup ◽  
Special Case ◽  
Geometric Action

Abstract.Given a complete CAT(0) space X endowed with a geometric action of a group Ⲅ, it is known that if Ⲅ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove the converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Ⲅ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

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.

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