On endomorphisms of torsion-free hyperbolic groups

Abstract

Let $H$ be a torsion-free $\delta$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a uniform conjugator if and only if $W(a_{1*},..., a_{n*})$ is conjugate to $W(a_1,..., a_n)$ for every word $W$ in $n$ variables and length up to a computable constant depending only on $\delta$, $\sharp{S}$ and $\sum_{i=1}^n |a_i|$. As a corollary, we deduce that there exists a computable constant $\mathcal{C}=\mathcal{C}(\delta, \sharp S)$ such that, for any endomorphism $\phi$ of $H$, if $\phi(h)$ is conjugate to $h$ for every element $h\in H$ of length up to $\mathcal {C}$, then $\phi$ is an inner automorphism. Another corollary is the following: if $H$ is a torsion-free conjugacy separable hyperbolic group, then $\text{\rm Out}(H)$ is residually finite. When particularizing the main result to the case of free groups, we obtain a solution for a mixed version of the classical Whitehead's algorithm.