Contraction groups in complete Kac-Moody groups (Q936139)

The main result of this paper is the following Theorem: Let \(G\) be an abstract Kac-Moody group over a finite field and \(\overline{G}\) be the closure of the image of \(G\) in the automorphism group of its positive building. Then the contraction group of any topologically periodic element in \(\overline{G}\) is trivial. Furthermore, if the type of \(G\) is irreducible and neither spherical nor affine, then the contraction groups of any element that is not topologically periodic in \(\overline{G}\) is not closed. Finally, the group \(\overline{G}\) contains non-topologically periodic elements whenever \(G\) is not of spherical type. As a consequence, all contraction groups of inner automorphisms are closed for geometric completions of Chevalley group schemes over the rings of Laurent polynomials over finite fields.