Fixed points for group actions on 2-dimensional affine buildings

Abstract

We prove a local-to-global result for fixed points of groups acting on affine buildings (possibly non-discrete) of types ${\tilde{A}}_2$ or ${\tilde{C}}_2$. In the discrete case, our theorem establishes the corresponding special cases of a conjecture by Marquis.

Substantial revision, generalised Busemann function argument to CAT(0) spaces, streamlined some other proofs and added explanation why our method does not seem to work for the ${\tilde{G}}_2$case