Some applications of ℓp-cohomology to boundaries of Gromov hyperbolic spaces

Marc Bourdon, Bruce Kleiner

Research output: Contribution to journalArticle

Abstract

We study quasi-isometry invariants of Gromov hyperbolic spaces, focusing on the ℓp -cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous ℓp-cohomology, thereby obtaining information about the ℓp-equivalence relation, as well as critical exponents associated with ℓp-cohomology. As an application, we provide a flexible construction of hyperbolic groups which do not have the Combinatorial Loewner Property, extending [8] and complementing the examples from [10]. Another consequence is the existence of hyperbolic groups with Sierpinski carpet boundary which have conformal dimension arbitrarily close to 1. In particular, we answer questions of Mario Bonk, Juha Heinonen and John Mackay.

Original languageEnglish (US)
Pages (from-to)435-478
Number of pages44
JournalGroups, Geometry, and Dynamics
Volume9
Issue number2
DOIs
StatePublished - Jan 1 2015

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Keywords

  • Asymptotic properties of groups
  • Cohomology of groups
  • Hyperbolic groups and nonpositively curved groups

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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