Explicit strong boundedness for higher rank symplectic groups

TL;DR

This paper provides an explicit, quantitative proof of strong boundedness for higher rank symplectic groups over rings of S-algebraic integers and semi-local rings, extending previous results to a broader class of groups.

Contribution

It introduces an explicit argument for strong boundedness of ${ m Sp}_{2n}(R)$, generalizing earlier results from ${ m SL}_n$ to symplectic groups and other split Chevalley groups.

Findings

Established explicit bounds for ${ m Sp}_{2n}(R)$

Extended strong boundedness results to semi-local rings

Generalized results to all split Chevalley groups

Abstract

This paper gives an explicit argument to show strong boundedness for ${\rm Sp}_{2n}(R)$ for $R$ a ring of S-algebraic integers or a semi-local ring. This gives a quantitative version of a related abstract result in a previous paper of the author. The results presented further generalize older results regarding strong boundedness by Kedra, Libman and Martin and Morris from ${\rm SL}_n$ to ${\rm Sp}_{2n}$. Further, the presented results solve the question of the asymptotic of strong boundedness for ${\rm Sp}_{2n}(R)$ for $R$ semi-local case with an argument that immediately generalizes to all other split Chevalley groups.