Abstract
We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then there is a global fixed point. In particular, all actions of finitely generated torsion groups on such complexes have global fixed points. The proofs rely on Masur’s theorem on periodic trajectories in rational billiards, and Ballmann–Brin’s methods for finding closed geodesics in 2-dimensional locally CAT(0) complexes. As another ingredient, we prove that the image of an immersed loop in a graph of girth 2π with length not commensurable with π has diameter >π. This is closely related to a theorem of Dehn on tiling rectangles by squares.
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