Horospherical limit points of S-arithmetic groups

Suppose Γ 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 Γ acts by isometries on a certain CAT(0) metric space XS = ∏ v∈S Xv, where each Xv is either a Euclidean building or a Riemannian symmetric space. For a point ξ on the visual boundary of XS , we show there exists a horoball based at ξ that is disjoint from some Γ-orbit in XS if and only if ξ lies on the boundary of a certain type of flat in XS 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 X.