Contraction groups in complete Kac–Moody groups

Abstract

Let G be an abstract Kac–Moody group over a finite field and G 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 G which are not topologically periodic are not closed. (In such groups there always exist elements that are not topologically periodic.)