THE MARTIN BOUNDARY ACTION OF GROMOV HYPERBOLIC COVERING GROUPS AND APPLICATIONS TO HARDY CLASSES

1995 ◽  
Vol 06 (04) ◽  
pp. 601-624 ◽  
Author(s):  
FINNUR LÁRUSSON
Keyword(s):  
Martin Boundary ◽  
Hardy Classes ◽  
Gromov Hyperbolic ◽  
Covering Groups

CHEEGER ISOPERIMETRIC CONSTANTS OF GROMOV-HYPERBOLIC SPACES WITH QUASI-POLES

2000 ◽  
Vol 02 (04) ◽  
pp. 511-533 ◽  
Author(s):  
JIANGUO CAO
Keyword(s):  
Martin Boundary ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Spaces ◽  
Connected Components ◽  
Local Geometry ◽  
Complete Manifold ◽  
Cheeger Constant ◽  
Gromov Hyperbolic ◽  
Geometric Boundary ◽  
Dirichlet Problem At Infinity

Let X be a non-compact complete manifold (or a graph) which admits a quasi-pole and has bounded local geometry. Suppose that X is Gromov-hyperbolic and the diameters (for a fixed Gromov metric) of the connected components of X(∞) have a positive lower bound. Under these assumptions we show that X has positive Cheeger isoperimetric constant. Examples are also constructed to show that the Cheeger constant h(X) may be zero if any of the above assumption on X is removed. Applications of this isoperimetric estimate include the solvability of the Dirichlet problem at infinity for non-compact Gromov-hyperbolic manifolds X above. In addition, we show that the Martin boundary ∂ΔX of such a space X is homeomorphic to the geometric boundary X(∞) of X at infinity.

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.

Martin boundary of unlimited covering surfaces

2000 ◽  
Vol 82 (1) ◽  
pp. 55-72 ◽  
Author(s):  
Hiroaki Masaoka ◽  
Shigeo Segawa
Keyword(s):  
Martin Boundary

scholarly journals Traces of certain classes of holomorphic functions on finite unions of Carleson sequences

1999 ◽  
Vol 41 (1) ◽  
pp. 103-114 ◽  
Author(s):  
ANDREAS HARTMANN
Keyword(s):  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions ◽  
Hardy Classes ◽  
Type Spaces ◽  
Trace Spaces

We give a method allowing the generalization of the description of trace spaces of certain classes of holomorphic functions on Carleson sequences to finite unions of Carleson sequences. We apply the result to different classes of spaces of holomorphic functions such as Hardy classes and Bergman type spaces.

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
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