Some applications of -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


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

Cite this article

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

DOI 10.4171/GGD/318