Hyperbolic buildings, conformal dimension and Mostow rigidity
type: Item
instance of: scholarly article
title: Hyperbolic buildings, conformal dimension and Mostow rigidity
zbMATH Open document ID: 0876.53020
DOI: 10.1007/PL00001619
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-11-02T00:00:00Z
review text: Let \((p,q)\) be such that \(p\geqslant 5\) and \(q\rightsquigarrow 2\), and \(I_{pq}\) be the Tits building whose apartments are the hyperbolic planes of curvature \(-1\), the chambers are the regular hyperbolic \(p\)-gones with angles of measure \( \pi \)/2 and whose link at each vertex is a complete bipartite \(q\)-graph. A graph is bipartite if its set of vertices can be partitioned into two disjoint subsets such that no two vertices in the same subset lie on a common edge. \(I_{pq}\) is a CAT(\(-1\)) space, which means that the geodesic triangles are thinner then their comparison triangles in the hyperbolic plane. Therefore it has a boundary at infinity. In this paper the author obtains a numerical expression that gives the exact conformal Pansu dimension of the boundary of the building. This was known in such an exact way only for homogeneous manifolds of negative curvature. The second theorem in this article shows that in this type of buildings the conformal dimension reflects the combinatorial structure of the building, and does not privilege any CAT(\(-1\)) structure, as is the case in symmetric non-compact spaces of rank one. In the second part of this theorem it is proved using Patterson-Sullivan measures, that if \(G\) is a cocompact lattice of \(I_{pq}\) such that \(q\geqslant 3\), and \(X\) is a CAT(\(-1\))-space on which \(G\) acts by isometries in a proper discontinuous way, then the conformal Pansu dimension of the boundary of the building is strictly smaller then the Hausdorff dimension of the boundary of \(X\). The third theorem shows that if \(q\geqslant 3\) and \(G\), \(G'\) are two isometric cocompact lattices from \(I_{pq}\) and \(I_{p'q' \text{ }}\), respectively, then \((p, q)=(p',q')\), and the isomorphism is the conjugate of an element of Isom (\(I_{pq}\)).To prove this last theorem the author uses an ergodic argument by \textit{D. Sullivan} [Bull. Am. Math. Soc., New Ser. 6, 57-73 (1982; Zbl 0489.58027)].
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1013772
zbMATH Keywords: hyperbolic buildings
zbMATH Keywords: Pansu's conformal dimension
zbMATH Keywords: Mostow rigidity
zbMATH Keywords: combinatorial Gromov product
zbMATH Keywords: Patterson-Sullivan metrics
author: Marc Bourdon
reviewed by: Ana Pereira do Vale
MaRDI profile type: Publication
full work available at URL: pl00001619
OpenAlex ID: W1982085400
description: scientific article; zbMATH DE number 1013772