A fixed point theorem for Lie groups acting on buildings and applications to Kac-Moody theory

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometrics on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac-Moody group are obtained by exponentialing ad-locally finite elements of the corresponding Lie algebra.