Metric compacti cations of locally symmetric spaces
نویسنده
چکیده
We introduce hyperbolic and asymptotic compactiications of metric spaces and apply them to locally symmetric spaces ?nX. We show that the reductive Borel{Serre compactiication ?nX RBS is hyperbolic and, as a corollary, get a result of Borel, and Kobayashi{Ochiai that the Baily{Borel compactiication ?nX BB is hyperbolic. We prove that the hyperbolic reduction of the toroidal compactiications ?nX tor is equal to ?nX BB and use it to derive a result of Kiernan{ Kobayashi on extensions of holomorphic maps from ?nX to the compactiication ?nX BB. We also show that the Tits compactiication ?nX T is an asymptotic compactiication while the asymptotic reduction of both ?nX BS and ?nX tor is equal to the end compactiication of ?nX, and prove a conjecture of Siegel on some metric properties of Siegel sets.
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