Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

TL;DR

This paper advances understanding of the geometric and algebraic properties of arithmetic groups by establishing polynomial isoperimetric inequalities and finiteness properties, especially for solvable subgroups within parabolic groups.

Contribution

It generalizes Bux's theorem to certain solvable groups and develops a field-independent reduction theory for arithmetic groups.

Findings

Polynomial isoperimetric inequalities for solvable subgroups

Finiteness properties of specific arithmetic groups

A field-independent approach to reduction theory

Abstract

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for the most part, independent of whether the underlying global field is a number field or a function field.