Hervé Pajot

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

In this paper we shall show that the boundary 9Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincare type inequalities. Then by using Heinonen-Koskelas work, we shall prove Loewner capacity estimates for some families of curves of WIp,q and the fact that every quasiconformal homeomorphism f : 0Ip,q d9Ip,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.

Quasi-Conformal Geometry and Hyperbolic Geometry

These notes deal with connections between quasi-conformal and hyperbolic geometry. In particular, we show how tools in geometric function theory like Poincare inequalities or Loewner spaces can be used to study problems in hyperbolic geometry, for instance the problem of rigidity of quasi-isometries in Gromov hyperbolic spaces.

The Geometric Traveling Salesman Problem in the Heisenberg Group

In the Heisenberg group H (endowed with its Carnot-Carathéodory structure), we prove that a compact set E ⊂ H which satisfies an analog of Peter Jones’ geometric lemma is contained in a rectifiable curve. This quantitative condition is given in terms of Heisenberg β numbers which measure how well the set E is approximated by Heisenberg straight lines.