Fixed points for group actions on 2-dimensional affine buildings (Q6663911)

The study of local-to-global results for fixed points of groups acting on affine buildings originated with \textit{J. P. Serre} (see [Trees. Transl. from the French by John Stillwell. Corrected 2nd printing of the 1980 original. Berlin: Springer (2003; Zbl 1013.20001)], section 6.5), who proved such a result for simplicial trees. Serre's result was extended to \(\mathbb {R}\)-trees by \textit{J. W. Morgan} and \textit{P. B. Shalen} in [Ann. Math. (2) 120, 401--476 (1984; Zbl 0583.57005)].\N\NIn the paper under review, the authors prove a local-to-global result for fixed points of finitely generated groups acting on \(2\)-dimensional affine buildings of types \(\widetilde{A}_{2}\) and \(\widetilde{C}_{2}\). The main result is Theorem A: Let \(G\) be a finitely generated group of automorphisms of a \(2\)-dimensional affine building \(X\) of type \(\widetilde{A}_{2}\) or \(\widetilde{C}_{2}\) (possibly nondiscrete). If every element of \(G\) fixes a point of \(X\), then \(G\) fixes a point of \(X\).\N\NThe proof combine building-theoretic arguments with standard results for CAT(0) spaces.