Some applications of ℓ<inf>p</inf>-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 ℓ<inf>p</inf> -cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous ℓ<inf>p</inf>-cohomology, thereby obtaining information about the ℓ<inf>p</inf>-equivalence relation, as well as critical exponents associated with ℓ<inf>p</inf>-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 - 2015

Fingerprint

Hyperbolic Space
Cohomology
Hyperbolic Groups
Quasi-isometry
Sierpinski Carpet
Invariant
Equivalence relation
Critical Exponents
Modulus

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

Some applications of ℓ<inf>p</inf>-cohomology to boundaries of Gromov hyperbolic spaces. / Bourdon, Marc; Kleiner, Bruce.

In: Groups, Geometry, and Dynamics, Vol. 9, No. 2, 2015, p. 435-478.

Research output: Contribution to journalArticle

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