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


Authors: Marc Bourdon and Hervé Pajot
Journal: Proc. Amer. Math. Soc. 127 (1999), 2315-2324
MSC (1991): Primary 30C65, 51E24
DOI: https://doi.org/10.1090/S0002-9939-99-04901-1
Published electronically: April 9, 1999
MathSciNet review: 1610912
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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 by M. Bourdon 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 quasisymmetric. Therefore by these results, the answer to questions 19 and 20 of Heinonen and Semmes (Thirty-three YES or NO questions about mappings, measures and metrics, Conform Geom. Dyn. 1 (1997), 1-12) is NO.


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Additional Information

Marc Bourdon
Affiliation: Institut Elie Cartan, Département de mathématiques, Université de Nancy I, BP 239, 54506 Vandoeuvre les Nancy, France
Email: marc.bourdon@iecn.u-nancy.fr

Hervé Pajot
Affiliation: Mathematical Science Research Institute, 1000 Centennial Drive, Berkeley, California 94720-5070
Address at time of publication: Département de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, BP222 Pontoise, 95302 Cergy-Pontoise Cédex, France
Email: pajot@u-cergy.fr

DOI: https://doi.org/10.1090/S0002-9939-99-04901-1
Keywords: Hyperbolic building, Poincar\'e inequality, quasiconformal mapping
Received by editor(s): October 28, 1997
Published electronically: April 9, 1999
Additional Notes: Parts of this work were done during a stay of the second author at MSRI. Research at MSRI is supported in part by NSF grant DMS-9022140.
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society