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

Abstract:

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 $\Delta$.