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We show, under mild hypotheses, that if each element of a finitely generatedgroup acting on a $2$-dimensional $\mathrm{CAT}(0)$ complex has a fixed point,then there is a global fixed point. In particular all actions of finitelygenerated torsion groups on such complexes have global fixed points. The proofsrely on Masur's theorem on periodic trajectories in rational billiards, andBallmann-Brin's methods for finding closed geodesics in $2$-dimensional locally$\mathrm{CAT}(0)$ complexes. As another ingredient we prove that the image ofan immersed loop in a graph of girth $2\pi$ with length not commensurable with$\pi$ has diameter $> \pi$. This is closely related to a theorem of Dehn ontiling rectangles by squares.

Torsion groups do not act on $2$-dimensional $\mathrm{CAT}(0)$ complexes