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 the action is trivial. In particular, all actions of finitely generated torsion groups on such complexes are trivial. As an ingredient, we prove that the image of an immersed loop in a graph of girth 2π with length not commensurable to π has diameter > π. This is related to a theorem of Dehn on tiling rectangles by squares.
This is joint work with Sergey Norin and Damian Osajda.[-]