The \(\ell_p\)-cohomology and the conformal dimension of hyperbolic cones
type: Item
instance of: scholarly article
title: The \(\ell_p\)-cohomology and the conformal dimension of hyperbolic cones
zbMATH Open document ID: 0899.53035
DOI: 10.1023/A:1004920322337
author: Gábor Elek
published in: Geometriae Dedicata
publication date: 1998-10-28T00:00:00Z
review text: Let \(X\) be a hyperbolic graph. One can define a constant called the critical \(l_{p}\)-cohomology dimension of \(X\), denoted by \(\text{clp} (X)\) and given by \(\inf\{p\mid H_{p}^{1}(X)\neq 0\}\), where \(H_{p}^{1}(X)\) is the first (nonreduced) \(l_{p}\)-cohomology of \(X\). \textit{M. Gromov} introduced in [`Geometric group theory. Volume 2: Asymptotic invariants of finite groups' (Lond. Math. Soc. Lect. Notes Ser. 182, Cambridge University Press) (1993; Zbl 0841.20039)] several quasi-isometric invariants for hyperbolic spaces, including the conformal dimension and the cut conformal dimension, which are both upper bounds for \(\text{clp} (X)\). All these invariants are related to a metric boundary of \(X\) called \(l_{p}\)-corona (also introduced by M. Gromov). The article begins with the description of the above notions as well as some of their properties. It is proved that quasi-isometric spaces have quasi-isometric \(l_{p}\)-coronas. Let \(K\) be a compact metric space; it is known that the Gromov boundary of the hyperbolic cone of \(K\) is homeomorphic to \(K\). For any compact space \(K\subset {\mathbb{R}}^{n}\), the author gives the construction of a hyperbolic graph \(C_{K}\) which is quasi-isometric to the hyperbolic cone. Moreover, the conformal dimension of \(C_{K}\) is at most the box dimension of \(K\).
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1121161
zbMATH Keywords: hyperbolic spaces
zbMATH Keywords: box dimension
zbMATH Keywords: quasi-isometric invariants
zbMATH Keywords: conformal dimension
zbMATH Keywords: \(l_p\)-corona
reviewed by: Ana Pereira do Vale
MaRDI profile type: Publication
full work available at URL: a:1004920322337
OpenAlex ID: W81307383
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description: scientific article; zbMATH DE number 1121161