Borel–Serre compactification

Borel–Serre compactification
Name	Borel–Serre compactification
Field	Algebraic topology; Differential geometry; Number theory
Introduced by	Armand Borel; Jean-Pierre Serre
Year	1973

Borel–Serre compactification. The Borel–Serre compactification is a construction for arithmetic locally symmetric spaces introduced by Armand Borel and Jean-Pierre Serre that produces a manifold with corners used in the study of cohomology of arithmetic groups and automorphic forms. It plays a central role in the work connecting Harish-Chandra's representation theory, the Langlands program, and results of Matsushima and Müller on analytic torsion, while interfacing with the theories of Eisenstein series, Hecke operators, and the Arthur trace formula.

Introduction

The construction applies to quotients of symmetric spaces associated to reductive algebraic groups such as GL_n, SL_n, Sp_n, and SO(p,q), and involves the action of arithmetic subgroups like SL_2(Z), GL_n(Z), or congruence subgroups of Adelic groups; it compactifies noncompact locally symmetric spaces in a way compatible with the boundary degenerations appearing in the work of Borel, Harish-Chandra, and Langlands. The compactification complements the work of Satake and Baily–Borel compactification by giving a topological and differential-geometric boundary stratified by rational parabolic subgroups studied by Tits and Bruhat–Tits theory; it has proved useful in the studies of Matsushima formula, Zucker conjecture, and the cohomological comparison theorems of Deligne and Weil.

Construction

Starting from a connected reductive group over Q such as GSp(2n), Res_(F/Q) GL_n instances, or split forms like SL_n over Q, one considers the symmetric space associated to the real Lie group studied by Élie Cartan and Helgason. The Borel–Serre compactification is obtained by adjoining boundary components indexed by conjugacy classes of rational parabolic subgroups classified via Levi decomposition and the Tits building studied by Jacques Tits, and by gluing partial compactifications of horospherical coordinates developed in the analytic approach of Harish-Chandra and Iwasawa decomposition. The resulting space is a manifold with corners whose strata correspond to flag varieties appearing in the work of Borel and Tits and to cusp degenerations examined in research by Selberg and Maaß.

Topological and Geometric Properties

Topologically the compactification yields a space homotopy equivalent to a finite CW-complex related to the Borel–Moore homology theories used by Borel and Moore, and it carries a natural stratification by faces indexed by parabolic conjugacy classes as in the theory of Weyl group actions and Bruhat decomposition. Geometrically, it admits a structure of a manifold with corners compatible with Riemannian metrics invariant under maximal compact subgroups as in the work of Cartan and Weyl, and it interacts with Hodge-theoretic degenerations studied by Deligne and Schmid in variations of Hodge structure associated to locally symmetric varieties such as those considered by Griffiths. The boundary behavior is crucial in analysis of spectral theory for Laplace-type operators following techniques of Atiyah–Patodi–Singer and Melrose.

Cohomological Applications

The compactification provides a framework to compute the cohomology of arithmetic groups like SL_n(Z), GL_2(Z), and congruence subgroups, connecting to the theory of automorphic forms developed by Langlands and calculations by Harder and Borel–Wallach; it makes precise the contribution of cuspidal and Eisenstein cohomology as studied by Franke and Zucker. It is used in proofs of the Eichler–Shimura relations and in the study of special values of L-functions investigated by Deligne and Beilinson, and it underlies comparison theorems between Betti, de Rham, and étale cohomology central to the work of Grothendieck and Fontaine. The Borel–Serre boundary also features in the analysis of the L^2-cohomology and in Zucker’s conjecture proven by Looijenga, Saper, and Stern in various cases.

Variants and Extensions

Several variants relate to other compactifications such as the Satake compactification, the Baily–Borel compactification for Hermitian locally symmetric domains studied by Baily and Borel, and partial compactifications used in the study of degenerations in arithmetic geometry by Mumford and Deligne–Mumford. Extensions include compatibility with the Adelic formulation of automorphic theory as in Weil and Tate, equivariant versions for actions of arithmetic Hecke algebras appearing in the work of Iwaniec and Jacquet–Langlands, and analytic refinements used in trace formula stabilization developed by Arthur and Kottwitz.

Examples and Computations

Basic examples include the Borel–Serre compactification of modular curves associated to SL_2(Z) and congruence subgroups where the boundary corresponds to cusps studied by Hecke and Atkin, and higher-rank examples for SL_3(Z), SL_4(Z), and Sp_4(Z) whose boundary stratifications were analyzed by Ash and Grayson. Explicit computations of cohomology with local coefficients have been carried out in cases treated by Harder, Sczech, and Emerton, and they connect to explicit constructions of Eisenstein cohomology and special value formulas studied by Shimura and Petersson. Deeper computational projects relating to torsion in homology and links with Galois representations are ongoing in the programs of Calegari–Geraghty and collaborations involving Harris and Taylor.

Category:Algebraic topology Category:Differential geometry Category:Number theory