Diophantine Equations and the Freeness of Möbius Groups

Abstract

Let p and q be two fixed no n zero integers verifying the condition gcd ( p , q ) = 1. We check solutions in non zero integers a 1 , b 1 , a 2 , b 2 and a 3 for the following Diophantine equations: ( B 1) ( B 2) . The equations ( B 1) and ( B 2) were considered by R.C. Lyndon and J.L. Ullman in [1] and A.F. Beardon in [2] in connection with the freeness of the M ? bius group generated by two matrices of namely and where . They proved that if one of the equations ( B 1) or ( B 2) has solutions in non zero integers then the group is not free. We give algorithms to decide if these equations admit solutions. We obtain an arithmetical criteria on p and q for which (B1) admits solutions. We show that for all p and q the equations ( B 1) and ( B 2) have only a finite number of solutions.