ANDREWS-CURTIS GROUPS

For any group G and integer k ≥ 2 the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group AC_k (G), on the subset N_k (G) ⊂ 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 AC_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 F AC(G) generated by AC-transformations on a much simpler set G^k. Our goal here is to investigate the natural epimorphism λ: F AC_k (G) → AC_k (G). We show that if G is non-elementary torsion-free hyperbolic, then F AC_k (G) acts faithfully on every nontrivial orbit of G^k, hence λ: F AC_k (G) → AC_k (G) is an isomorphism.