THE Σ-INVARIANTS OF S-ARITHMETIC SUBGROUPS OF BOREL GROUPS

Given a Chevalley group g of classical type and a Borel subgroup 13 subset of g, we compute the Sigma-invariants of the S-arithmetic groups 13(Z[1/N]), where N is a product of large enough primes. To this end, we let 13(Z[1/N]) act on a Euclidean building X that is given by the product of Bruhat-Tits buildings Xp associated to g, where p is a prime dividing N. In the course of the proof we introduce necessary and sufficient conditions for convex functions on CAT(0)-spaces to be continuous. We apply these conditions to associate to each simplex at infinity Tau subset of partial differential infinity X its so-called parabolic building X Tau and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential n-connectivity rather than actual n-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building Delta contains an apartment, provided Delta is thick enough and Aut(Delta) acts chamber transitively on Delta.