The \(\ell_p\)-cohomology and the conformal dimension of hyperbolic cones
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 \(K\).
scientific article; zbMATH DE number 1121161