VISUAL LIMITS OF MAXIMAL FLATS IN SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS

Abstract

Let X be a Riemannian symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let ∂∞X denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building ∂∞X: this defines a natural, geometric compactification of the space of maximal flats of X. We then completely determine the possible degenerations of apartments when X is of rank 1 or associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit remarkable behaviours of visual limits of maximal flats in various symmetric spaces of small rank and the surprising algebraic restrictions that occur.