Reduction theory using semistability. (English) Zbl 0564.20027

Grayson, Daniel R.

Comment. Math. Helv. 59, 600-634 (1984).

This paper gives a fresh approach in some cases to the Borel-Serre compactification of locally symmetric spaces of arithmetic type. The method begins from a construction of Stuhler’s of a canonical filtration of a given lattice L in Euclidean space. The elements in the filtration are sublattices K that give vertices of the convex hull of the set of points (dim K, log volume K) as K runs through all sublattices of L. This set is bounded in the (x,y) plane. Functions which measure the distance from L to a given ”cusp” at infinity are given in terms of these canonical filtrations. Deleting all L’s closer than a certain fixed amount to any cusp, we get a contractible manifold M whose boundary has the homotopy type of the appropriate Tits building.
If A is the ring of integers in a number field F and P a finitely generated projective A-module, we take for our L’s the group P equipped with an inner product on \(P\otimes F_ v\) for every infinite place v. Then GL(P) will act properly discontinuously on M with compact quotient and all the results of A. Borel and J. P. Serre [in Comment. Math. Helv. 48, 436-491 (1973; Zbl 0274.22011)] are reproven for the arithmetic groups commensurate with GL(P). In the last section, the methods and results are extended to orthogonal and symplectic groups. This paper is reasonably self-contained.

Reviewer: A.Ash

MSC:

20H05	Unimodular groups, congruence subgroups (group-theoretic aspects)
11E04	Quadratic forms over general fields
20G10	Cohomology theory for linear algebraic groups

Keywords:

Borel-Serre compactification of locally symmetric spaces; sublattices; cusp; contractible manifold; homotopy type; Tits building; symplectic groups

Citations:

Zbl 0274.22011