POINCARÉ INEQUALITIES AND QUASICONFORMAL STRUCTURE ON THE BOUNDARY OF SOME HYPERBOLIC

BUILDINGS

MARC BOURDON AND HERVÉ PAJOT

Abstract. In this paper we shall show that the boundary ∂I_{p,q} of the hyperbolic building I_{p,q} considered in M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow (Geometric And Functional Analysis, Vol 7 (1997), p 245-268) admits Poincaré type inequalities. Then by using Heinonen-Koskela’s work, we shall prove Loewner capacity estimates for some families of curves of ∂I_{p,q} and the fact that every quasiconformal homeomorphism f : ∂I_{p,q} ⟶ ∂I_{p,q} is quasisymetric. Therefore by these results, the answers to certain questions of Heinonen and Semmes are NO.