New York Journal of Mathematics
Volume 20 (2014) 353-366


Grigori Avramidi and Dave Witte Morris

Horospherical limit points of finite-volume locally symmetric spaces

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Published: April 10, 2014
Keywords: Horospherical limit point, locally symmetric space, Tits building, arithmetic group, Ratner's theorem
Subject: 53C35 (Primary); 20G30, 22E40 (Secondary)

Suppose X/Γ is an arithmetic locally symmetric space of noncompact type (with the natural metric induced by the Killing form of the isometry group of X), and let ξ be a point on the visual boundary of X. T.Hattori showed that if each horoball based at ξ intersects every Γ-orbit in X, then ξ is not on the boundary of any Q-split flat in X. We prove the converse. (This was conjectured by W.H.Rehn in some special cases.) Furthermore, we prove an analogous result when Γ is a nonarithmetic lattice.

Author information

Grigori Avramidi:
Department of Mathematics, University of Chicago, Chicago, IL 60637
Current Address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090

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