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

TL;DR

This paper computes the $\Sigma$-invariants of $S$-arithmetic subgroups of Borel groups in Chevalley groups, using geometric and combinatorial techniques involving buildings and convex functions on CAT(0)-spaces.

Contribution

It introduces new conditions for convex functions on CAT(0)-spaces, applies them to associate parabolic buildings to simplices at infinity, and develops novel combinatorial Morse theory methods.

Findings

Computed $\Sigma$-invariants for specific $S$-arithmetic groups

Established new conditions for convex functions on CAT(0)-spaces

Proved the existence of apartments opposite chambers in thick spherical buildings

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 $p$ runs over the primes dividing $N$. In the course of the proof we introduce necessary and sufficient conditions for convex functions on $\mbox{CAT(0)}$-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 $X^{\tau}$, which we study 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 $\mbox{Aut}(\Delta)$ acts chamber transitively on $\Delta$.