ON ENDOMORPHISMS OF TORSION-FREE HYPERBOLIC GROUPS

Abstract

Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. From the main result in the paper, Theorem 1.2, we deduce the following two corollaries. First, we show that there exists a computable constant [Formula: see text] such that, for any endomorphism φ of H, if φ(h) is conjugate to h for every element h ∈ H of length up to [Formula: see text], then φ is an inner automorphism. Second, we show a mixed (conjugate/non-conjugate) version of the classical Whitehead problem for tuples is solvable in torsion-free hyperbolic groups.