Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings

Abstract

1 ) Any quasi-conformal homeomorphism arises as the boundary homeomorphism of a quasi-isometry by [Tuk].( 2 ) The boundary of CH^ can be endowed with an Ison^CH") invariant contact structure by projecting the contact structure from a unit tangent sphere SJ"" 1 CH" to BCH^* using the exponential map.Corollary 1.1.4(Quasi-isometric classification of symmetric spaces).-Let X, X' be symmetric spaces of noncompact type.If X and X' are quasi-isometric, then they become isometric after the metrics on their de Rham factors are suitably renormalized.Mostow's work [Mos] implies that two quasi-isometric rank 1 symmetric spaces of noncompact type are actually isometric (up to a scale factor); and it was known by [AS] that two quasi-isometric symmetric spaces of noncompact type have the same rank.We will discuss other applications of Theorems 1 1.2 and 1.1.3elsewhere, see [KlLe2] and [KlLe3].( 1 ) The distance function on the product space is given by the Pythagorean formula.