## Journal of Differential Geometry

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

### Compactifications of Locally Symmetric Spaces

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 27 April 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1146169912

**Digital Object Identifier**

doi:10.4310/jdg/1146169912

**Mathematical Reviews number (MathSciNet)**

MR2226955

**Zentralblatt MATH identifier**

1122.22005

#### Citation

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