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

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

1997-10-20

arxiv.org/abs/math/9710208v1

Mathematics

Differential Geometry

Scientific paper

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 rigidit\'e de Mostow} (Geometric And Functional Analysis, Vol 7 (1997), p 245-268) admits Poincar\'e 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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