Equations in the Q–completion of a Torsion–free Hyperbolic Group

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group. It was recently proved in [1] that finite systems of equations over the Q–completion of a finitely generated free group are algorithmically solvable. In this paper we generalize the results of [1] to the case of the Q–completion G of an arbitrary torsion–free hyperbolic group G with generators d1, . . . , dN . (A detailed definition of the Q–completion of a group can be found in [1].) Main Theorem. Let G be a torsion–free hyperbolic group. Then there exists an algorithm that decides if a given finite system of equations over the Q–completion of G has a solution, and if it does, finds a solution. By a triangular equation we mean an equation with at most three terms. Given a system S of equations in G, we may assume that all equations in S are triangular. Indeed, an arbitrary equation x1 1 x ε2 2 . . . x εn n = 1 (where xi stands either for a variable or for a constant, εi = ±1) is equivalent to a finite system of triangular equations: x1 1 x ε2 2 y −1 1 = 1, y1x ε3 3 y −1 2 = 1, . . . , yn−2x εn−1 n−1 x εn n = 1. Adding a finite number of new variables and equations, we may also assume that S contains only equations with coefficients in G. To achieve this, we replace every constant of the form d m n by a new variable z satisfying the equation z = d. From now on, we fix a finite system S of triangular equations over G with coefficients in G. We will reduce the system S to a finite set of systems in a specific hyperbolic group G ∗ 〈t1, . . . , tψ(m)〉, where the number ψ(m) can be determined effectively given the number m of equations in the original system S. The resulting systems are accompanied by the restriction that some of the variables belong to a certain subgroup G ∗ 〈t1, . . . , ti〉 of G ∗ 〈t1, . . . , tψ(m)〉, i.e. do not contain certain t’s. To each of the systems we can apply the (slightly modified) method of Rips and Sela [4] to see if they are decidable. If none of them is consistent, then the original system S has no solution; if at least one has a solution, then it is possible to find a corresponding solution to the system S. Suppose that the system has a solution in G; let {X1, . . . , XL} be a solution with the minimal possible number of roots. Since the solution belongs to G, it is contained in a certain group K, obtained from G by adding finitely many roots. It is the union of a chain of subgroups Hi defined as follows. Let G = H0. Received by the editors August 14, 1996. 1991 Mathematics Subject Classification. Primary 20E05, 20F10. The first author was supported by grants from NSERC and FCAR. The third author was supported by the NSF grant DMS-9103098. c ©1999 American Mathematical Society