JournalsggdVol. 9, No. 2pp. 435–478

Some applications of p\ell_p-cohomology to boundaries of Gromov hyperbolic spaces

  • Marc Bourdon

    Université Lille I, Villeneuve d'Ascq, France
  • Bruce Kleiner

    Courant Institute of Mathematical Sciences, New York, United States
Some applications of $\ell_p$-cohomology to boundaries of Gromov hyperbolic spaces cover

Abstract

We study quasi-isometry invariants of Gromov hyperbolic spaces, focusing on the p\ell_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous p\ell_p-cohomology, thereby obtaining information about the p\ell_p-equivalence\linebreak relation, as well as critical exponents associated with p\ell_p-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 11. In particular, we answer questions of Mario Bonk, Juha Heinonen and John Mackay.

Cite this article

Marc Bourdon, Bruce Kleiner, Some applications of p\ell_p-cohomology to boundaries of Gromov hyperbolic spaces. Groups Geom. Dyn. 9 (2015), no. 2, pp. 435–478

DOI 10.4171/GGD/318