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 e2 2 . x en n = 1 (where xi stands either for a variable or for a constant, ei = ±1) is equivalent to a finite system of triangular equations: x1 1 x e2 2 y −1 1 = 1, y1x e3 3 y −1 2 = 1, . , yn−2x en−1 n−1 x en 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 2961 License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use 2962 O. KHARLAMPOVICH, E. LIOUTIKOVA, AND A. MYASNIKOV Step 1. Consider pairwise nonconjugated cyclically minimal primitive elements u1, . , uk1 ∈ G, |u1| ≤ . ≤ |uk1 | (here |u| denote the length of u in G), and add roots t1, . , tk1 , such that uj = t sj j . (Notice that ui+1 does not become a proper power after we add roots t1, . . . , ti.) The corresponding groups are denoted by H1, . , Hk1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step 2. Consider pairwise nonconjugated primitive elements uk1+1, . , uk2 ∈ H1, cyclically reduced in the amalgamated product, each having the reduced form u = t1 1 c1 . t αk 1 ck, where αi 6= 0, αi ∈ Z, ci ∈ G, |uk1+1|H1 ≤ . ≤ |uk2 |H1 ; and add roots tk1+1, . , tk2 , such that uj = t sj j , to the group Hk1 . The corresponding groups are denoted by Hk1+1, . , Hk2 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step i+1. Suppose that H1, . , Hki have been constructed. Consider pairwise nonconjugated primitive elements uki+1, . , uki+1 ∈ Hi, cyclically reduced in the amalgamated product, each having the reduced form t1 i c1 . tk i ck, where αi 6= 0, αi ∈ Z, ci ∈ Hi−1 ( ck is not a power of ui, because the elements are cyclically reduced), |uki+1|Hi ≤ . ≤ |uki+1 |Hi , and add roots tki+1, . , tki+1 , such that uj = t sj j , to the group Hki . The corresponding groups are denoted by Hki+1, . , Hki+1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Finally, for some number i one has K = Hki+1 . The group Hki+1 is called the group at level i, corresponding to the sequence u1, . , uki+1 . The group Hi will be called the group of rank i. We also order the set of tj ’s: tk < tl if k < l. Let Fi be the free group with the same generating set as Hi, i = 1, . , k1, and let β : Hi → Fi be a section, i.e. a mapping of sets such that π ◦ β = id. For our purposes, β has to be of specific form; we will define it explicitly in section 2. For X ∈ Hi we will call β(X) the canonical representative of X . Definition 1. By an equational diagram over a group Hi we mean an equation X1 . Xn = 1 together with a solution A1, . , An and a diagram over Hi having β(A1) . β(An) as its boundary label. An equational triangle is an equational diagram corresponding to a triangular equation. A system of equational diagrams is a system of equations together with the system of diagrams such that the solution associated to each equational diagram is a solution of the whole system. A free equational diagram is an equational diagram with no cells; the corresponding equations are called free equations. If the level i of the group Hki+1 is greater than zero, we can use the method of [1] to reduce each equation of the system S to a bounded number of free equations and at most one equation over the group in the previous rank. We can proceed in the same way until we reach level 0 (i.e. the group Hk1), for which we need to develop a separate method. In what follows we will denote Hk1 by H . 1. Some properties of the Cayley graph of H We will frequently use the following lemma: Lemma 1. ([3, Lemma 1.5]) For each geodesic triangle [x1, x2, x3] in a δ-space, there are points yi ∈ [xi−1, xi+1] (indices are considered modulo 3) such that d(xi, yi−1) = d(xi, yi+1) = (xi−1 · xi+1)xi , License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use THE Q–COMPLETION OF A TORSION–FREE HYPERBOLIC GROUP 2963