A reduction point algorithm for cocompact Fuchsian groups and applications

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May 2014, 8(2): 223-239. doi: 10.3934/amc.2014.8.223

A reduction point algorithm for cocompact Fuchsian groups and applications

Pilar Bayer ^1, and Dionís Remón ^1,

1.	Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain, Spain

Received October 2013 Published May 2014

Abstract
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In the present article we propose a reduction point algorithm for any Fuchsian group in the absence of parabolic transformations. We extend to this setting classical algorithms for Fuchsian groups with parabolic transformations, such as the flip flop algorithm known for the modular group $\mathbf{SL}(2, \mathbb{Z})$ and whose roots go back to [9]. The research has been partially motivated by the need to design more efficient codes for wireless transmission data and for the study of Maass waveforms under a computational point of view.

Keywords: fundamental domains, Fuchsian codes, SISO channels., reduction point algorithm, Quaternionic Fuchsian groups.

Mathematics Subject Classification: 11F06, 11Y16, 14G50, 30F35, 65Y04, 94B4.

Citation: Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223

References:

[1]	M. Alsina and P. Bayer, Quaternion Orders, Quadratic Forms and Shimura Curves, Amer. Math. Soc., Providence, 2004.
[2]	P. Bayer, Contributions to Shimura curves, in Win-Women in Numbers: Research Directions in Number Theory, 2011, 15-33.
[3]	I. Blanco-Chacón, C. Hollanti and D. Remón, Fuchsian codes for AWGN channels, in International Workshop on Coding and Cryptography 2013, Bergen, 496-507.
[4]	E. Brandani da Silva, M. Firer, S. Costa and R. Palazzo, Signal constellations in the hyperbolic plane: a proposal for new communication systems, J. Franklin Institute, 343 (2006), 69-82. doi: 10.1016/j.jfranklin.2005.09.001.
[5]	E. D. Carvalho, A. A. Andrade, R. Palazzo and J. Vieira, Arithmetic Fuchsian groups and space time codes, Comput. Appl. Math., 30 (2011), 485-498. doi: 10.1590/S1807-03022011000300001.
[6]	D. Hejhal and B. Rackner, On the topography of Maass waveforms for PSL(2,Z), Exp. Math., 1 (1992), 275-305.
[7]	S. Katok, Fuchsian Groups, Univ. Chicago Press, 1992.
[8]	A. Lascurain, Some presentations for $\overline{\Gamma}_0(N)$, Conform. Geom. Dyn., 6 (2002), 33-60. doi: 10.1090/S1088-4173-02-00073-5.
[9]	J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973.
[10]	F. Strömberg, Maass waveforms on $(\Gamma_0(N), \chi)$ (computational aspects), in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology (eds. J. Bolt and F. Steiner), Cambridge Univ. Press, 2012, 187-228.
[11]	J. Voight, Computing fundamental domains for Fuchsian groups, J. Théor. Nombres Bordeaux, 21 (2009), 467-489.

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References:

[1]	M. Alsina and P. Bayer, Quaternion Orders, Quadratic Forms and Shimura Curves, Amer. Math. Soc., Providence, 2004.
[2]	P. Bayer, Contributions to Shimura curves, in Win-Women in Numbers: Research Directions in Number Theory, 2011, 15-33.
[3]	I. Blanco-Chacón, C. Hollanti and D. Remón, Fuchsian codes for AWGN channels, in International Workshop on Coding and Cryptography 2013, Bergen, 496-507.
[4]	E. Brandani da Silva, M. Firer, S. Costa and R. Palazzo, Signal constellations in the hyperbolic plane: a proposal for new communication systems, J. Franklin Institute, 343 (2006), 69-82. doi: 10.1016/j.jfranklin.2005.09.001.
[5]	E. D. Carvalho, A. A. Andrade, R. Palazzo and J. Vieira, Arithmetic Fuchsian groups and space time codes, Comput. Appl. Math., 30 (2011), 485-498. doi: 10.1590/S1807-03022011000300001.
[6]	D. Hejhal and B. Rackner, On the topography of Maass waveforms for PSL(2,Z), Exp. Math., 1 (1992), 275-305.
[7]	S. Katok, Fuchsian Groups, Univ. Chicago Press, 1992.
[8]	A. Lascurain, Some presentations for $\overline{\Gamma}_0(N)$, Conform. Geom. Dyn., 6 (2002), 33-60. doi: 10.1090/S1088-4173-02-00073-5.
[9]	J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973.
[10]	F. Strömberg, Maass waveforms on $(\Gamma_0(N), \chi)$ (computational aspects), in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology (eds. J. Bolt and F. Steiner), Cambridge Univ. Press, 2012, 187-228.
[11]	J. Voight, Computing fundamental domains for Fuchsian groups, J. Théor. Nombres Bordeaux, 21 (2009), 467-489.

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