Bounded Conjugators For Real Hyperbolic and Unipotent Elements in Semisimple Lie Groups

TL;DR

This paper establishes bounds on conjugating elements in real semisimple Lie groups for hyperbolic and unipotent elements, showing that these bounds can often be uniform, which advances understanding of conjugation dynamics in such groups.

Contribution

It provides explicit bounds on conjugators for hyperbolic and unipotent elements in real semisimple Lie groups, including conditions for uniform bounds.

Findings

Bound on conjugating elements depends on properties of the elements

For most elements, a uniform bound is applicable

Improves understanding of conjugation in semisimple Lie groups

Abstract

Let $G$ be a real semisimple Lie group with trivial centre and no compact factors. Given a conjugate pair of either real hyperbolic elements or unipotent elements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that $d_G(1,g) \leq L(d_G(1,u)+d_G(1,v))$, where $L$ is a positive constant which will depend on some property of $a$ and $b$. For the vast majority of such elements however, $L$ can be assumed to be a uniform constant.