Locally elliptic actions, torsion groups, and nonpositively curved spaces

Thomas Haettel, Damian Osajda

Published 2021-10-24, updated 2022-02-24Version 2

Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic (that is, every element has a bounded orbit) actions by automorphisms of finitely generated groups on finite dimensional nonpositively curved complexes have global fixed points. In particular, finitely generated torsion groups cannot act without fixed points on such spaces. We prove these conjectures for a wide class of complexes, including all infinite families of Euclidean buildings, Helly complexes, some graphical small cancellation and systolic complexes, uniformly locally finite Gromov hyperbolic graphs. We present consequences of these result, e.g. concerning the automatic continuity. Our main tool are Helly graphs. On the way we prove several results concerning their automorphisms. This is of independent interest and includes a classification result: any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic, with rational translation length. One consequence is that groups with distorted elements cannot act properly on such graphs. We also present and study a new notion of geodesic clique paths. Their local-to-global properties are crucial in our proof of ellipticity results.