Decompositions of congruence subgroups of Chevalley groups

TL;DR

This paper extends classical decompositions of Chevalley groups over rings, providing new insights and bounds on subgroup widths, with results applicable even in finite field cases.

Contribution

It formulates and proves relative decompositions for Chevalley groups over rings, offering new bounds on principal congruence subgroup widths and extending known results.

Findings

New relative decompositions for Chevalley groups over rings

Upper bounds for principal congruence subgroup widths

Results applicable to finite fields and rings

Abstract

We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in terms of several families of generators. Some of our results are new even in the absolute case and were previously studied only for groups over finite fields.