PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 3, July 1980 TRACE POLYNOMIAL FOR TWO GENERATOR SUBGROUPSOF SL(2,Q CHARLES R. TRAINA1 Abstract. If G is a group generated by two 2x2 matrices A and B having determinant +1, with entries from the complex field C, it is known that the trace of any word in A and B, W(A, B) is a polynomial with integral coefficients in the three variables: x = traced), >> = trace(fi), z — tracers), defined as trace W(A, B) = P(x,y, z), where P is determined uniquely by the conjugacy class of W(A, B). The actual computation of this trace polynomial is not easily obtained. It is the purpose of this paper to derive an explicit formula for this trace polynomial, and to indicate some consequences of it. 1. Introduction. Fricke, in his work on automorphic functions [1] had shown the following: Let G be a group generated by finitely many 2 by 2 matrices with entries from the field C of complex numbers and with determinant +1. Then the trace of any word in the generators is a polynomial with integral coefficients in finitely many variables which are the traces of the generators and of finitely many of their products. In particular, if G has two generators A and B, then the trace of any word W(A, B) is a polynomial in x,y, z where x = traced, v = trace B, z = trace AB. (1) Horowitz [2] has shown that x,y,z can be considered as independent variables and that for every triplet of values x0, y^, z0 there exists a pair of unimodular matrices A0, B0 such that trace A0 = x0, trace B0 = y0 and trace AqB0 = z0. In addition, A0, B0 are, in general, i.e. whenever xl + yl + z% - x0yoz0 - 4 ¥=0, (2) uniquely determined up to conjugacy within the linear group SL(2, Q. Since the group generated by two matrices A, B is free of rank 2 unless x,y, z satisfy at least one of a countable number of algebraic equations with integral coefficients, it follows that every conjugacy class of an element W(a, b) in a free group on free generators a, b determines uniquely a polynomial P(x, y, z) with integral coeffi- cients if we define P by trace W(A, B) = P(x,y, z). (3) Received by the editors November 13, 1978. AMS (MOS) subject classifications (1970). Primary 20F10; Secondary 20C20. 'Taken from the Representation of the trace polynomial of cyclically reduced words in a free group on two generators submitted to the Faculty of the Polytechnic Institute of New York in partial fulfillment of the requirements for the degree Doctor of Philosophy (Mathematics), 1978. © 1980 American Mathematical Society 0002-9939/80/0000-030S/$02.00 369 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use