Conjugacy width in uniform higher rank arithmetic groups of orthogonal type
Abstract
We study widths of conjugacy classes in anisotropic higher rank -arithmetic groups of orthogonal type. Assuming the GRH, we prove that many such groups have bounded conjugacy width. For example, this holds if the degree is greater or equal to 17 and contains a non-archimedean place. To the best of our knowledge, this is the first boundedness result proved for anisotropic groups. The proof uses ideas from the Congruence Subgroup Problem. In particular, we define and compute a non standard version of the metaplectic kernel. Conversely, we prove that a quantitative bound on the width of conjugacy classes implies the CSP. The machinery we develop can also be used for other width questions. For example, in \cite{AM25} we prove, unconditional on GRH, new cases of bounded generation of arithmetic groups.
Keywords
Cite
@article{arxiv.2304.13173,
title = {Conjugacy width in uniform higher rank arithmetic groups of orthogonal type},
author = {Nir Avni and Chen Meiri},
journal= {arXiv preprint arXiv:2304.13173},
year = {2025}
}
Comments
expanded version, now covering many orthogonal groups