Andrews-Curtis groups

For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $A{C}_k(G)$, on the subset $N_k(G)\subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $A{C}_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $\lambda \text{\hspace{0.17em}⁣}:FA{C}_k(G)\to A{C}_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FA{C}_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $\lambda \text{\hspace{0.17em}⁣}:FA{C}_k(G)\to A{C}_k(G)$ is an isomorphism.