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Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings

Marc Bourdon·Hervé Pajot
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Abstract

In this paper we shall show that the boundary $\partial I_{p,q}$ of the hyperbolic building $I_{p,q}$ considered in M. Bourdon, \emph{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 $\partial I_{p,q}$ and the fact that every quasiconformal homeomorphism $f : \partial I_{p,q} \longrightarrow \partial I_{p,q}$ is quasisymetric. Therefore by these results, the answers to certain questions of Heinonen and Semmes are NO.

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Marc Bourdon, Hervé Pajot. 1997-10-20. Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. https://arxiv.org/abs/math/9710208

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