Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

By M. Bonk, Bruce Kleiner. 2002

118 citations recorded by the source corpus.

Suppose G is a Gromov hyperbolic group, and @1G is quasisymmetrically homeomorphic to an Ahlfors Q–regular metric 2–sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H 3 .

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1377b07b9ead7c6e573c9d5cef7b6e67ff517ffb

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On the conformal gauge of a compact metric space (2012, methodology): “We note the importance of this fact in the proof of the theorem of Bonk and Kleiner [BonK05].”
Uniformization of two-dimensional metric surfaces (2014, background): “ly if Y is linearly locally contractible. This is a geometric condition which in particular rules out cusp-like behavior, see Section 16. This result has been extended in several consequent works cf. [12], [40], [59], [60]. There the Ahlfors regularity condition is combined with varying geometric co”
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity (2006, background): “Here a weak tangent is a pointed Gromov–Hausdorff limit of a sequence of rescalings of Z, see [42], [6].”
Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups (2010, methodology): “By imitating an argument from [BK05a], one shows that the CLP holds provided it holds for pairs of balls (Proposition 2.”
The Poincare inequality is an open ended condition (2008, methodology): “This result was used by Bonk and Kleiner ([4]) who subsequently showed that such metric measure spaces that arise as the boundary of a Gromov hyperbolic group, are Loewner.”
On the Assouad dimension of self-similar sets with overlaps (2014, background): “Here the conformal dimension of X is the infimum of the dimension of quasi-conformal images of X. Keith and Laakso [15] study Ahlfors regular metric spaces whose Assouad dimension cannot be lowered by quasi-conformal maps and Bonk and Kleiner [4] apply this work to resolve Cannon’s conjecture in cer”