Abstract
We introduce hyperbolic and asymptotic compactifications of metric spaces and apply them to locally symmetric spaces Γ\X. We show that the reductive Borel–Serre compactification [Formula: see text] is hyperbolic and, as a corollary, get a result of Borel, and Kobayashi–Ochiai that the Baily–Borel compactification [Formula: see text] is hyperbolic. We prove that the hyperbolic reduction of the toroidal compactifications [Formula: see text] is equal to [Formula: see text] and use it to derive a result of Kiernan–Kobayashi on extensions of holomorphic maps from Γ\ X to the compactification [Formula: see text]. We also show that the Tits compactification [Formula: see text] is an asymptotic compactification while the asymptotic reduction of both [Formula: see text] and [Formula: see text] is equal to the end compactification Γ\ X, and prove a conjecture of Siegel on some metric properties of Siegel sets.
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