Groups, Geometry, and Dynamics
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Published online: 2015-06-08
Some applications of $\ell_p$-cohomology to boundaries of Gromov hyperbolic spacesMarc Bourdon and Bruce Kleiner (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  and complementing the examples from . 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