Abstract
Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic actions (that is, every element has a bounded orbit) of finitely generated groups on finite dimensional nonpositively curved spaces 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 spaces, including all infinite families of Euclidean buildings, Helly complexes, some graphical small cancellation and systolic complexes, uniformly locally finite Gromov hyperbolic graphs. We present numerous consequences of these result, e.g. concerning the automatic continuity. On the way we prove several results concerning automorphisms of Helly graphs. They are of independent interest and include 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.