The Σ-invariants of 𝑆-arithmetic subgroups of Borel groups

Abstract

Given a Chevalley group G \mathcal {G} of classical type and a Borel subgroup B ⊆ G \mathcal {B} \subseteq \mathcal {G} , we compute the Σ \Sigma -invariants of the S S -arithmetic groups B ( Z [ 1 / N ] ) \mathcal {B}(\mathbb {Z}[1/N]) , where N N is a product of large enough primes. To this end, we let B ( Z [ 1 / N ] ) \mathcal {B}(\mathbb {Z}[1/N]) act on a Euclidean building X X that is given by the product of Bruhat–Tits buildings X p X_p associated to G \mathcal {G} , where p p is a prime dividing N N . In the course of the proof we introduce necessary and sufficient conditions for convex functions on C A T ( 0 ) CAT(0) -spaces to be continuous. We apply these conditions to associate to each simplex at infinity τ ⊂ ∂ ∞ X \tau \subset \partial _\infty X its so-called parabolic building X τ 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 n -connectivity rather than actual n 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 Δ \Delta contains an apartment, provided Δ \Delta is thick enough and A u t ( Δ ) Aut(\Delta ) acts chamber transitively on Δ \Delta .