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

Abstract

We show, under mild hypotheses, that if each element of a finitely generated group 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 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 $\mathrm{CAT}(0)$ complexes. As another ingredient we prove that the image of an 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 on tiling rectangles by squares.

v3, 27 pages, section 5 corrected, final version accepted for publication

References in corpus (1)

• Presqu'un immeuble pour le groupe des automorphismes modérés

Cited by in corpus (1)

• Tits Alternative for groups acting properly on $2$-dimensional recurrent complexes (with an appendix written jointly with Jon McCammond)