Some applications of ${\ell }_p$-cohomology to boundaries of Gromov hyperbolic spaces

Abstract

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.