Dimension conforme et sphère à l'infini des variétés à courbure négative

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

“…2 The proof which we present here is independent of [39]. We avoid the analytic issue entirely by relying instead on combinatorial arguments using a generalized definition of the conformal modulus first introduced by Pansu in [51] and developed in [69]. 3 Thus even in the Euclidean setting our approach provides a new proof for Gehring's classical theorem.…”

“…This concept was originally introduced by Pansu [51], [52] in his study of Lipschitz and quasiconformal maps on the boundaries of rank one symmetric spaces. Pansu's definition was used in [69] to study quasisymmetry and geometric quasiconformality in general Ahlfors regular spaces.…”

“…The classical proof of the celebrated Mostow rigidity theorem required the development of an analytic theory of quasiconformal mappings on the boundaries at infinity of the classical rank one symmetric spaces. Quasiconformal maps in the setting of nonabelian Carnot groups were first studied by Mostow and later by Pansu [51], [52] and Korányi and Reimann [42], [43]. Until recently, however, it was not entirely clear to what extent the existence of this non-Euclidean theory was dependent on the inherent algebraic structure of these spaces.…”

“…For this purpose, we introduce a generalized modulus based on packingtype measures. Generalized moduli based on Hausdorff-type measures were used in [51], [52] and [69].…”

Happy fractals and some aspects of analysis on metric spaces

“…Hyperbolic groups have very interesting spaces at infinity associated to them which satisfy the doubling property described in the next section. In addition to the references already mentioned, see [19,41,70,71] in this regard. Note that fundamental groups of compact Riemannian manifolds without boundary and with strictly negative sectional curvatures are nonelementary hyperbolic groups.…”