Combinatorial Modulus on Boundary ofRight-Angled Hyperbolic Buildings

2014/11/30 by Antoine Clais
Mathematics · #Boundary (topology) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Modulus #math.GR #math.GT #msc:20E42 #msc:20F67 #msc:20F69

paper · pdf · doi:10.1515/agms-2016-0001

published as Analysis and Geometry in Metric Spaces, Volume 4, Issue 1, Pages 1-53, 2016 · Revised version after referee comments

arxiv created 2015/10/25 · openalex publication_date 2016/02/11 · arxiv updated 2016/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Abstract In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D.Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity of right-angled Fuchsian buildings.

Citations

• Lectures on Analysis on Metric Spaces
• Sur les Groupes Hyperboliques d’après Mikhael Gromov
• Quasisymmetric parametrizations of two-dimensional metric spheres
• The combinatorial Riemann mapping theorem
• Cut points and canonical splittings of hyperbolic groups
• Quasiconformal maps in metric spaces with controlled geometry
• Recognizing constant curvature discrete groups in dimension 3
• Conformal dimension and Gromov hyperbolic groups with 2–sphere boundary