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Extending and unifying a number of well-known conjectures and open questions,we conjecture that locally elliptic (that is, every element has a boundedorbit) actions by automorphisms of finitely generated groups on finitedimensional nonpositively curved complexes have global fixed points. Inparticular, finitely generated torsion groups cannot act without fixed pointson such spaces. We prove these conjectures for a wide class of complexes,including all infinite families of Euclidean buildings, Helly complexes, somegraphical small cancellation and systolic complexes, uniformly locally finiteGromov 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 resultsconcerning their automorphisms. This is of independent interest and includes aclassification result: any automorphism of a Helly graph with finitecombinatorial dimension is either elliptic or hyperbolic, with rationaltranslation length. One consequence is that groups with distorted elementscannot act properly on such graphs. We also present and study a new notion ofgeodesic clique paths. Their local-to-global properties are crucial in ourproof of ellipticity results.

Locally elliptic actions, torsion groups, and nonpositively curved spaces