Poincar\'e inequalities and quasiconformal structure on the boundary of some hyperbolic buildings

In this paper we shall show that the boundary $\mathrm{\partial }{I}_{p,q}$ 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 $\mathrm{\partial }{I}_{p,q}$ 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.