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Commutator subgroups of the power subgroups of generalized Hecke groups. (English) Zbl 1448.20042

Summary: Let \(p, q\geqslant 2\) be relatively prime integers and let \(H_{p,q}\) be the generalized Hecke group associated to \(p\) and \(q\). The generalized Hecke group \(H_{p,q}\) is generated by \(X(z)=-(z-\lambda_p)^{-1}\) and \(Y(z)=-(z+\lambda_q)^{-1}\) where \(\lambda_p=2\cos \frac{\pi}{p}\) and \(\lambda_q=2\cos \frac{\pi}{q} \). In this paper, for positive integer \(m\), we study the commutator subgroups \((H_{p,q}^m)'\) of the power subgroups \(H_{p,q}^m\) of generalized Hecke groups \(H_{p,q} \). We give an application related with the derived series for all triangle groups of the form \((0;p,q,n)\), for distinct primes \(p, q\) and for positive integer \(n\).

MSC:

20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
11F06 Structure of modular groups and generalizations; arithmetic groups
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[1] F. Ates, I.N.Cangül, E.K. Cetinalp, A.S. Cevik and E. G. Karpuz,On Commutator and Power Subgroups of Some Coxeter Groups, Appl. Math. Inf. Sci. 10, No. 2,
[2] F. Ates and A.S. Cevik,Knit products of some groups and their applications,Rend. Semin. Mat. Univ. Padova, 121, 1-11 (2009). · Zbl 1186.20022
[3] I. I. Bouw and M. Möller,Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172, no. 1, 139-185 (2010). · Zbl 1203.37049
[4] G. Burde and H. Zieschang,Knots, Second edition. De Gruyter Studies in Mathematics, 5. Walter de Gruyter & Co., Berlin, 2003. · Zbl 1009.57003
[5] I. N. Cangül, R. Sahin, S. Ikikardes and Ö. Koruoğlu,Power subgroups of some Hecke groups. II., Houston J. Math.33, no. 1, 33-42. · Zbl 1122.20024
[6] I. N. Cangül and D. Singerman,Normal subgroups of Hecke groups and regular maps, Math. Proc. Camb. Phil. Soc.123, 59-74 (1998). · Zbl 0893.20036
[7] K. Calta and T. A. Schmidt,Infinitely many lattice surfaces with special pseudoAnosov maps, J. Mod. Dyn. 7, No. 2, 239-254 (2013). · Zbl 1322.30015
[8] K. Calta and T. A. Schmidt,Continued fractions for a class of triangle groups, J. Aust. Math. Soc. 93, No. 1-2, 21-42 (2012). · Zbl 1336.11051
[9] A.S. Cevik, N.Y. Özgür, R. Sahin,The extended Hecke groups as semi-direct products and related results,Int. J. Appl. Math. Stat., 13: 63-72 (2008).
[10] B. Demir, Ö. Koruoğlu and R. Sahin,Conjugacy Classes of Extended Generalized Hecke Groups,Rev. Un. Mat. Argentina, 57, No. 1, 49-56 (2016). · Zbl 1345.20067
[11] B. Demir, Ö. Koruoğlu and R. Sahin,On Normal Subgroups of Generalized Hecke Groups,An. Şt. Univ. Ovidius Constanta, 24, No. 2, (2016). · Zbl 1389.20061
[12] E. Hecke,Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann.112, 664-699 (1936). · Zbl 0014.01601
[13] W. P. Hooper,Grid graphs and lattice surfaces, Int. Math. Res. Not. IMRN , no. 12, 2657-2698 (2013). · Zbl 1333.37047
[14] S. Huang,Generalized Hecke groups and Hecke polygons, Ann. Acad. Sci. Fenn., Math. 24, No.1, 187-214 (1999). · Zbl 0926.30026
[15] S. Ikikardes, O. Koruoglu and R. Sahin,Power subgroups groups of some Hecke groups,Rocky Mountain Journal of Mathematics, No. 2, (2006). · Zbl 1179.20044
[16] Ş. Kaymak, B. Demir, Ö. Koruoğlu and R. Sahin,Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups,An. Ştiinţ. Univ. · Zbl 1438.20045
[17] C. L. Lang and M. L. Lang,Arithmetic and geometry of the Hecke groups, J. Algebra 460, 392-417 (2016). · Zbl 1417.11032
[18] C. L. Lang and M. L. Lang,Identifying normal and congruence subgroups, http://arxiv.org/pdf/1501.00743.pdf. · Zbl 1129.81059
[19] J. Lehner,Uniqueness of a class of Fuchsian groups,Illinois J. Math. 19, 308-315 (1975). · Zbl 0305.20024
[20] J. Lehner, and M. Newman,Real two-dimensional representations of the modular group and related groups, Amer. J. Math. 87, 945-954 (1965). · Zbl 0138.31803
[21] G. J. Martin,The geometry and arithmetic of Kleinian groups. Handbook of group actions,Vol. I, 411-494, Adv. Lect. Math. (ALM), 31, Int. Press, Somerville, MA,
[22] H. Movasati and K. M. Shokri,Automorphic forms for triangle groups: integrality properties, J. Number Theory 145, 67-78 (2014). · Zbl 1297.11039
[23] L. P. Neuwirth,A remark on knot groups with a center, Proc. Amer. Math. Soc. 14, 378-379 (1963). · Zbl 0118.39303
[24] M. Newman and J. R. Smart,Note on a subgroup of the modular group, Proc. Amer. Math. Soc. 14, 102-104 (1963). · Zbl 0113.02803
[25] J. Nielsen,Kommutatorgruppen für das freie Produkt von zyklischen Gruppen. (Danish), Mat. Tidsskr. B, 49-56 (1948). · Zbl 0033.09902
[26] S. Nugent and J. Voight,On the arithmetic dimension of triangle groups,Math. Comp. 86, no. 306, 1979-2004 (2017). · Zbl 1379.11043
[27] R. Sahin and O. Bizim,Some subgroups of the extended Hecke groupsH(λq), Acta Math. Sci., Ser. B, Engl. Ed.23, No.4, 497-502 (2003). · Zbl 1042.20040
[28] R. Sahin, O. Bizim, and I. N. Cangül,Commutator subgroups of the extended Hecke groups, Czech. Math. J.54, No.1, 253-259 (2004). · Zbl 1053.11038
[29] R. Sahin, S. Ikikardes,Squares of congruence subgroups of the extended modular group, Miskolc Math. Notes, 14, 1031-1035 (2013). · Zbl 1286.11051
[30] R. Sahin, S. Ikikardes, and Ö. Koruoğlu,Some normal subgroups of the extended Hecke groupsH(λp), Rocky Mountain J. Math.36, no. 3, 1033-1048 (2006). · Zbl 1139.20042
[31] R. Sahin and Ö. Koruoğlu,Commutator Subgroups of the Power Subgroups of Hecke Groups H(λq),Ramanujan J., 24, no. 2, 151-159 (2011). · Zbl 1218.20033
[32] R. Sahin and Ö. Koruoğlu,Commutator subgroups of the power subgroups of Hecke groups H(λq) II, C. R. Math. Acad. Sci. Paris 349, no. 3-4, 127-130 (2011). · Zbl 1209.20041
[33] R. Sahin, Ö. Koruoğlu and S. Ikikardes,On the extended Hecke groupsH(λ5), Algebra Colloq., 13, 17-23 (2006). · Zbl 1088.20028
[34] R. Sahin, T. Meral and Ö. Koruoglu,Power and Free Normal Subgroups of Generalized Hecke Groups,
[35] Z. Sarıgedik, S. Ikikardes and R. Sahin,Power subgroups of the extended Hecke groups, Miskolc Math. Notes 16, no. 1, 483-490 (2015). · Zbl 1340.20045
[36] V. V. Tsanov,Triangle groups, automorphic forms, and torus knots, Enseign. Math. (2) 59, no. 1-2, 73-113 (2013). · Zbl 1301.57009
[37] W. A. Veech,Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97, no. 3, 553-583 (1989). · Zbl 0676.32006
[38] C. C. Ward,Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems 18, no. 4, 1019-1042 (1998). · Zbl 0915.58059
[39] R.
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