  We present a new operation to be performed on elements in a Garside group,
called cyclic sliding, which is introduced to replace the well known cycling
and decycling operations. Cyclic sliding appears to be a more natural choice,
simplifying the algorithms concerning conjugacy in Garside groups and having
nicer theoretical properties. We show, in particular, that if a super summit
element has conjugates which are 'rigid' (that is, which have a certain
particularly simple structure), then the optimal way of obtaining such a rigid
conjugate through conjugation by positive elements is given by iterated cyclic
sliding.
