  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.
