Diophantine Equations and the Freeness of Möbius Groups

Marin Gutan

Laboratoire de Mathématiques, Université Blaise-Pascal, Clermont-Ferrand, France.

DOI: 10.4236/am.2014.510132                

Abstract

Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a₁,b₁,a₂,b₂ and a₃ for the following Diophantine equations: (B1) (B2) .
The equations (B1) and (B2) 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 (B1) or (B2) 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 (B1) and (B2) have only a finite number of solutions.

Keywords

Diophantine Equation, Möbius Groups, Free Group

Conflicts of Interest

The authors declare no conflicts of interest.

References

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