Characters of free groups represented in the two‐dimensional special linear group

Abstract

We consider here the problem of determining when two elements in a free group will have the same character under all possible representations of the given group in the special linear group of 2 x 2 matrices with determinant 1. Problems involving the representation of free groups in terms of 2 x 2 matrices with real entries and determinant 1 have been investigated by R. Fricke in connection with certain problems in the theory of Riemann surfaces [2]. Fricke’s main concern is to find those representations which define discontinuous groups of the conformal self-mappings of the upper half-plane. His method is based on arguments in non-Euclidean geometry, and has only recently been made fully rigorous and transparent by Linda Keen [4]. Apart from the analytic problems inherent in Fricke’s method there are several algebraic problems which have been answered by Fricke either incompletely or not at all. It is the algebraic aspects of Fricke’s theory which we shall investigate here. Independently of Fricke’s problems however, the subsequent development may be considered as a straightforward contribution to the representation theory of free groups. We shall consider the following general situation. K will denote a fixed commutative ring with identity of characteristic zero, and SL(2, K) the group of all 2 x 2 matrices with determinant 1 and entries from K. It will be shown that the character of any element in a free group under an arbitrary representation of the group in SL(2, K) can be constructively represented as a polynomial expression with integer coefficients in the characters of certain simple products of the group generators. Two polynomials will represent the same character if and only if they are congruent modulo a member of a certain class of ideals. This will reduce the problem of determining whether two elements have the same character to the problem of determining the structure of these ideals. The structure of the ideals will then be determined for the free groups of rank 1, 2, and for the free group of rank 3 under restricted conditions on K. We shall show that if two elements u, v in a free group F have the same character under all possible * The work for this paper was done while the author was at the Courant Institute of Mathematical Sciences and was supported by the National Science Foundation Grant GP-28538. Reproduction in whole or in part is permitted for any purpose of the United States Government. 635