The Finitary Andrews-Curtis Conjecture

The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked in [5] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [5]. 1 Andrews-Curtis graphs Let G be a group and G be the set of all k-tuples of elements of G. The following transformations of the set G are called elementary Nielsen transformations (or moves): (1) (x1, . . . , xi, . . . , xk) −→ (x1, . . . , xix j , . . . , xk), i 6= j; (2) (x1, . . . , xi, . . . , xk) −→ (x1, . . . , x±1 j xi, . . . , xk), i 6= j; (3) (x1, . . . , xi, . . . , xk) −→ (x1, . . . , x−1 i , . . . , xk). Elementary Nielsen moves transform generating tuples of G into generating tuples. These moves together with the transformations (4) (x1, . . . , xi, . . . , xk) −→ (x1, . . . , xi , . . . , xk), w ∈ S ∪ S−1 ⊂ G, where S is a fixed subset of G, form a set of elementary Andrews-Curtis transformations relative to S (or, shortly, ACS-moves). If S = G then AC-moves transform n-generating tuples (i.e., tuples which generate G as a normal subgroup) into n-generating tuples. We say that two k-tuples U and V are ACSequivalent, and write U ∼S V , if there is a finite sequence of ACS-moves which transforms U into V . Clearly, ∼S is an equivalence relation on the set G of