New York Journal of Mathematics
Volume 20 (2014) 367-376


Dave Witte Morris and Kevin Wortman

Horospherical limit points of S-arithmetic groups

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Published: April 10, 2012
Keywords: Horospherical limit point, S-arithmetic group, Tits building, Ratner's theorem
Subject: 20G30 (Primary) 20E42, 22E40, 51E24 (Secondary)

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.

Author information

Dave Witte Morris:
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K6R4, Canada

Kevin Wortman:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090