METRIC COMPACTIFICATIONS OF LOCALLY SYMMETRIC SPACES

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.