Uniform boundedness for algebraic groups and Lie groups

TL;DR

This paper investigates boundedness properties of certain subgroups of algebraic and Lie groups, establishing conditions under which boundedness implies uniform boundedness and providing explicit bounds related to the group's rank.

Contribution

It proves that boundedness of the subgroup generated by minimal parabolic subgroups implies uniform boundedness and offers explicit bounds for these groups over algebraically closed and split fields.

Findings

If $G^+(k)$ is bounded, then it is uniformly bounded.

Explicit bounds for $ ext{diam}(G^+(k))$ are given: $ ext{diam}ig(G^+(k)ig) ext{leq}4 imes ext{rank} imes G$ over algebraically closed fields.

For split groups, the bound is $ ext{diam}ig(G^+(k)ig) ext{leq}28 imes ext{rank} imes G$.

Abstract

Let $G$ be a semisimple linear algebraic group over a field $k$ and let $G^+(k)$ be the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all the minimal $k$-parabolic subgroups $Q$ of $G$. We prove that if $G^+(k)$ is bounded then it is uniformly bounded. Under extra assumptions we get explicit bounds for $\Delta(G^+(k))$: we prove that if $k$ is algebraically closed then $\Delta(G^+(k))\leq 4\, {\rm rank}\,G$, and if $G$ is split over $k$ then $\Delta(G^+(k))\leq 28\, {\rm rank}\,G$. We deduce some analogous results for real and complex semisimple Lie groups.