Contraction groups in complete Kac-Moody groups

TL;DR

This paper investigates contraction groups in complete Kac-Moody groups over finite fields, showing that for certain irreducible, non-spherical, non-affine types, non-topologically periodic elements have non-closed contraction groups.

Contribution

It establishes a new property of contraction groups in these Kac-Moody groups, specifically their non-closure for non-topologically periodic elements in certain types.

Findings

Non-topologically periodic elements have non-closed contraction groups.

Existence of non-topologically periodic elements in these groups.

Contraction group properties depend on the Dynkin diagram type.

Abstract

Let $G$ be an abstract Kac-Moody group over a finite field and $\bar{G}$ be the closure of the image of $G$ in the automorphism group of its positive building. We show that if the Dynkin diagram associated to $G$ is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in $\bar{G}$ which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)