On Endomorphisms of torsion-Free hyperbolic Groups

Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. Let g1, . . . , gn and g1∗, . . . , gn∗ be elements of H such that gr∗ is conjugate to gr for each r = 1, . . . , n. There is a uniform conjugator if and only if W (g1∗, . . . , gn∗) is conjugate to W (g1, . . . , gn) for every word W in n variables and length up to a computable constant depending only on δ, ♯S and ∑ n r=1 |gr|. As a corollary we deduce, that there exists a computable constant C = C(δ, ♯S) such that for any endomorphism φ of H if φ(h) is conjugate to h for every element h ∈ H of length up to C, then φ is an inner automorphism. Another corollary is the following: if H is a torsion-free conjugacy separable hyperbolic group, then the group Out(H) is residually finite.