  A group with a geometric action on some hyperbolic space is necessarily word
hyperbolic, but on the other hand every countable group acts (metrically)
properly by isometries on a locally finite hyperbolic graph. In this paper we
consider what happens when a group acts isometrically on a restricted class of
hyperbolic spaces, for instance quasitrees. We obtain strong conclusions on the
group structure if the action has a locally finite orbit, especially if the
group is finitely generated.
  We also look at group actions on finite products of quasitrees, where our
actions may be by automorphisms or by isometries, including the Leary -
Minasyan group.
