The conjugacy problem in hyperbolic groups for finite lists of group   elements
  Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. A (non-practical) algorithm is described that, given as input two lists A and B, each composed of m words in the generators and their inverses, determines whether or not the lists are conjugate in G, and returns a conjugating element should one exist. The algorithm runs in time O(m mu)$, where mu is an upper bound on the lengths of elements in the two lists. Similarly, an algorithm is outlined that computes generators of the centraliser of A, with the same bound on running time. 
David J. Buckley, Derek F. Holt
math.GR