Groups, Geometry, and Dynamics

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Volume 9, Issue 2, 2015, pp. 435–478
DOI: 10.4171/GGD/318

Published online: 2015-06-08

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

Marc Bourdon[1] and Bruce Kleiner[2]

(1) Université Lille I, Villeneuve d'Ascq, France
(2) Courant Institute of Mathematical Sciences, New York, United States

We study quasi-isometry invariants of Gromov hyperbolic spaces, focusing on the $\ell_p$-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous $\ell_p$-cohomology, thereby obtaining information about the $\ell_p$-equivalence\linebreak relation, as well as critical exponents associated with $\ell_p$-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 $1$. In particular, we answer questions of Mario Bonk, Juha Heinonen and John Mackay.

Keywords: Hyperbolic groups and nonpositively curved groups, asymptotic properties of groups, cohomology of groups

Bourdon Marc, Kleiner Bruce: Some applications of $\ell_p$-cohomology to boundaries of Gromov hyperbolic spaces. Groups Geom. Dyn. 9 (2015), 435-478. doi: 10.4171/GGD/318