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Reduction theory using semistability. II. (English) Zbl 0616.20020
This paper extends the results of Part I [ibid. 59, 600-634 (1984; Zbl 0564.20027)] to the case of an arithmetic subgroup of any semisimple algebraic group G. The methods and results of the previous paper are assumed in the current one. A numerical notion of semistability of lattices endowed with metrics is used to construct distance functions to the cusps in the symmetric space X belonging to G, leading to a realization of the Borel-Serre completion of X as a proper subset of X. Of independent interest: (1) the description of X as a certain space of inner products H on the Lie algebra g of G; (2) a criterion for when an element of the canonical filtration of the $${\mathbb{Z}}$$-points of g (with respect to a given H) is a parabolic subalgebra of g, solely in terms of the slopes of the canonical filtration.
Reviewer: A.Ash

##### MSC:
 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects) 20G10 Cohomology theory for linear algebraic groups 11E04 Quadratic forms over general fields 20G15 Linear algebraic groups over arbitrary fields 11F06 Structure of modular groups and generalizations; arithmetic groups 11E57 Classical groups
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