Diophantine Equations and the Freeness of Möbius Groups

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 *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.

Let

The equations (

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.

Gutan, M. (2014) Diophantine Equations and the Freeness of Möbius Groups.

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*SL*_{2}(*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

[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

[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