Journal of Differential Geometry

Compactifications of Locally Symmetric Spaces

Armand Borel and Lizhen Ji

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Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a natural homogeneous space, whose quotient on the right by a maximal compact subgroup K of G gives a locally symmetric space Γ\G/K. This paper constructs several new compactifications of Γ\G. The first two are related to the Borel-Serre compactification and the reductive Borel-Serre compactification of the locally symmetric space Γ\G/K; in fact, they give rise to alternative constructions of these known compactifications. More importantly, the compactifications of Γ\G imply extension to the compactifications of homogeneous bundles on Γ\G/K, and quotients of these compactifications under non-maximal compact subgroups H provide compactifications of period domains Γ\G/H in the theory of variation of Hodge structures. Another compactification of Γ\G is obtained via embedding into the space of closed subgroups of G and is closely related to the constant term of automorhpic forms, in particular Eisenstein series.

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J. Differential Geom., Volume 73, Number 2 (2006), 263-317.

First available in Project Euclid: 27 April 2006

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Borel, Armand; Ji, Lizhen. Compactifications of Locally Symmetric Spaces. J. Differential Geom. 73 (2006), no. 2, 263--317. doi:10.4310/jdg/1146169912.

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