Topological simplicity, commensurator super-rigidity and nonlinearities of Kac-Moody groups (Q1762661)

This paper is written by the first author and contains an appendix written by the second author. The paper describes a new point of view on topological Kac-Moody groups as generalized semisimple groups over local fields, namely it is shown that topological Kac-Moody groups are products of topologically simple groups, and that the Iwahori subgroups of topological Kac-Moody groups are the normalizers of the pro-\(p\) Sylow subgroups. Using the dynamical characterization of parabolic subgroups it is shown that some countable Kac-Moody groups with Fuchsian buildings are not linear. Furthermore it is shown that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group. For the last statement the first author uses the commensurator super-rigidity theorem proved in the appendix by the second author.