The divergence of the special linear group over a function ring

TL;DR

This paper calculates the divergence of the special linear group over a function ring and shows that its asymptotic cones lack cut-points, revealing geometric properties of these algebraic groups.

Contribution

It provides the first explicit computation of divergence for SLn over function rings and links this to the topology of its asymptotic cones.

Findings

Divergence of SLn(O_S) is computed explicitly.

All asymptotic cones of the group have no cut-points.

Abstract

We compute the divergence of the finitely generated group SLn(O_S), where S is a finite set of valuations of a function field, and O_S is the corresponding ring of S-integer points. As an application, we deduce that all its asymptotic cones are without cut-points.