Connectivity Properties of Horospheres in Euclidean Buildings and Applications to Finiteness Properties of Discrete Groups

TL;DR

This paper links the local ranks of algebraic groups over global function fields to the finiteness properties of their S-arithmetic subgroups, using geometric analysis of horospheres in Euclidean buildings.

Contribution

It establishes a precise connection between local ranks and homological finiteness properties for rank 1 groups, confirming the sharpness of previous bounds.

Findings

Finiteness length of G(O_S) depends on sum of local ranks when K-rank is 1

Connectivity properties of horospheres in Euclidean buildings are characterized

The upper bound for finiteness length is shown to be sharp in this case

Abstract

Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S) established in an earlier paper is sharp in this case. The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.