Bounded generation by root elements for Chevalley groups defined over rings of integers of function fields with an application in strong boundedness

TL;DR

This paper proves that certain algebraic groups over rings of integers in function fields can be generated by root elements and satisfy strong boundedness, extending known results from number fields to function fields.

Contribution

It establishes bounded generation by root elements for Chevalley groups over rings of integers in global function fields, using model theoretic methods.

Findings

Proves bounded generation by root elements for these groups.

Demonstrates that these groups satisfy strong boundedness property.

Extends results from number fields to function fields.

Abstract

Bounded generation by root elements is a property which has been widely studied for various types of linear algebraic groups defined over rings of integers in algebraic number fields. However, when considering global function fields, there are not many results beyond the treatment of special cases due to Nica and Queen. In this paper, we use model theoretic methods due to Carter, Keller and Paige written up by Morris to prove bounded generation by root elements for simply connected, split Chevalley groups defined over the ring of all integers in a global function field. We further apply this bounded generation result together with results from a previous paper by the author to derive that the aforementioned Chevalley groups satisfy the strong boundedness property introduced by Kedra, Libman and Martin.