The \Sigma-invariants of S-arithmetic subgroups of Borel groups.

Given a Chevalley group \mathcal {G} of classical type and a Borel subgroup \mathcal {B} \subseteq \mathcal {G}, we compute the \Sigma-invariants of the S-arithmetic groups \mathcal {B}(\mathbb {Z}[1/N]), where N is a product of large enough primes. To this end, we let \mathcal {B}(\mathbb {Z}[1/N]) act on a Euclidean building X that is given by the product of Bruhat–Tits buildings X_p associated to \mathcal {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 \partial _\infty 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. [ABSTRACT FROM AUTHOR]