Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem

TL;DR

This paper establishes the precise finiteness length of S-arithmetic subgroups in isotropic reductive groups over global function fields, confirming conjectured properties using Euclidean building structures.

Contribution

It proves the exact finiteness length for these groups, extending understanding of their algebraic and geometric properties in positive characteristic.

Findings

Finiteness length equals sum of local ranks minus one.

Confirmed conjectured finiteness properties for isotropic reductive groups.

Utilized Behr-Harder reduction theory and Euclidean buildings.

Abstract

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.