Describing the nub in maximal Kac-Moody groups

TL;DR

This paper provides a complete description of the nub for any element in maximal Kac-Moody groups over finite fields, advancing the understanding of contraction groups in these complex tdlc groups.

Contribution

It offers the first comprehensive characterization of nubs in maximal Kac-Moody groups, a significant step in analyzing their structure.

Findings

Explicit description of the nub for all elements in these groups

Insights into when contraction groups are closed or not

Enhanced understanding of the structure of maximal Kac-Moody groups

Abstract

Let $G$ be a totally disconnected locally compact (tdlc) group. The contraction group $\mathrm{con}(g)$ of an element $g\in G$ is the set of all $h\in G$ such that $g^n h g^{-n} \to 1_G$ as $n \to \infty$. The nub of $g$ can then be characterized as the intersection $\mathrm{nub}(g)$ of the closures of $\mathrm{con}(g)$ and $\mathrm{con}(g^{-1})$. Contraction groups and nubs provide important tools in the study of the structure of tdlc groups, as already evidenced in the work of G. Willis. It is known that $\mathrm{nub}(g) = \{1\}$ if and only if $\mathrm{con}(g)$ is closed. In general, contraction groups are not closed and computing the nub is typically a challenging problem. Maximal Kac-Moody groups over finite fields form a prominent family of non-discrete compactly generated simple tdlc groups. In this paper we give a complete description of the nub of any element in these groups.