Horospherical limit points of S-arithmetic groups

Abstract

Suppose Gamma is an S-arithmetic subgroup of a connected, semisimple algebraic group G over a global field Q (of any characteristic). It is well known that Gamma acts by isometries on a certain CAT(0) metric space X_S that is a Cartesian product of Euclidean buildings and Riemannian symmetric spaces. For a point p on the visual boundary of X_S, we show there exists a horoball based at p that is disjoint from some Gamma-orbit in X_S if and only if p lies on the boundary of a certain type of flat in X_S that we call "Q-good." This generalizes a theorem of G.Avramidi and D.W.Morris that characterizes the horospherical limit points for the action of an arithmetic group on its associated symmetric space.