AM  Vol.5 No.10 , June 2014
Diophantine Equations and the Freeness of Möbius Groups
Author(s) Marin Gutan*
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.

Cite this paper
Gutan, M. (2014) Diophantine Equations and the Freeness of Möbius Groups. Applied Mathematics, 5, 1400-1411. doi: 10.4236/am.2014.510132.
References
[1]   Lyndon, R.C. and Ullman, J.L. (1969) Groups Generated by Two Linear Parabolic Transformations. Canadian Journal of Mathematics, 21, 1388-1403. http://dx.doi.org/10.4153/CJM-1969-153-1

[2]   Beardon, A.F. (1993) Pell’s Equation and Two Generator MÖbius Groups. Bulletin of the London Mathematical Society, 25, 527-532. http://dx.doi.org/10.1112/blms/25.6.527

[3]   Klarner, D., Birget, J.-C. and Satterfield, W. (1991) On the Undecidability of the Freeness of Integer Matrix Semigroups. International Journal of Algebra and Computation, 1, 223-226.
http://dx.doi.org/10.1142/S0218196791000146

[4]   Cassaigne, J., Harju, T. and Karhumaki, J. (1999) On the Undecidability of the Freeness of Matrix Semigroups. International Journal of Algebra and Computation, 9, 295-305.
http://dx.doi.org/10.1142/S0218196799000199

[5]   Cassaigne, J. and Nicolas, F. (2012) On the Decidability of Semigroup Freeness. RAIRO—Theoretical Informatics and Applications, 46, 355-399. http://dx.doi.org/10.1051/ita/2012010

[6]   Gawrychowski, P., Gutan, M. and Kisielewicz, A. (2010) On the Problem of Freness of Multplicative Matrix Semigroups. Theoretical Computer Science, 411, 1115-1120.
http://dx.doi.org/10.1016/j.tcs.2009.12.005

[7]   Farbman, S.P. (1995) Non-Free Two-Generator Subgroups of SL2(Q) . Publicacions Mathemàtiques, 39, 379-391. http://dx.doi.org/10.5565/PUBLMAT_39295_13

[8]   Tan, E.-C. and Tan, S.-P. (1996) Quadratic Diophantine Equations and Two Generators MÖbius Groups. Journal of the Australian Mathematical Society, 61, 360-368.
http://dx.doi.org/10.1017/S1446788700000434

[9]   de la Harpe, P. (2000) Topics in Geometric Group Theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago.

[10]   Grytczuk, A. and Wojtowicz, M. (2000) Beardon’s Diophantine Equations and Non-Free MÖbius Groups. Bulletin of the London Mathematical Society, 32, 305-310.
http://dx.doi.org/10.1017/S1446788700000434

[11]   Bamberg, J. (2000) Non-Free Points for Groups Generated by a Pair of 2 × 2 Matrices. Journal of the London Mathematical Society, 62, 795-801. http://dx.doi.org/10.1112/S0024610700001630

 
 
Top