Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators

Binlin Dai; Zekun Li

doi:doi:10.11648/j.acm.20221101.13

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Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators

Binlin Dai, Zekun Li

Published in Applied and Computational Mathematics (Volume 11, Issue 1)

Received: 18 January 2022 Accepted: 7 February 2022 Published: 19 February 2022

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Abstract

Let be real Möbius groups. For any 2×2 matrix A in induces real Möbius transformations g by the formula where . The collection of all real Möbius transformations for which takes the values 1 forms a group which can be identified with . We write to mean that f is a random variable in. In this paper, we study the random Möbius subgroup. We can get some new results: (1) If, the probability is greater than 0.027282. (2) If f is hyperbolic transformation and g is parabolic transformation, then the probability is discrete is greater than 7/12. (3) If f is elliptic of order n and g is elliptic of order 2, then the probability is discrete is greater than 2/n. (4) The probability that random chosen generate an elementary or non-discrete groupis greater than 0.0302049.

Published in	Applied and Computational Mathematics (Volume 11, Issue 1)
DOI	10.11648/j.acm.20221101.13
Page(s)	31-37
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Random Discrete Möbius Group, , Jørgensen’s Inequality

References

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[2]	A. F. Bardon. “The Geometry of Discrete Groups”, Graduate Text in Mathematics 91., Springer-Verlag, New York, 1983.
[3]	C. Cao. “Some trace inequalities for discrete groups of Möbius transformations”, Proc. Amer. Math. Soc., 123 (1995), 168-172.
[4]	J. D. Dixon. “The probability of generating the symmetric group”, Math. Z., 110 (1969), 119-205.
[5]	B. L. Dai, A. Fang and B. Nai. “Discreteness criteria for subgroups in complex hyperbolic space”, Proc. Japan Acad. Ser. A Math. Sci., 77 (2001), 168-172.
[6]	B. L. Dai, S. Xie. “on algebraic limits and topological properties of set of Möbius groups in ”, Complex Variables., 47 (2002), 149-154.
[7]	B. L. Dai. “Discreteness criteria for subgroups of isometry groups of pinched Hadamard Manifolds by use a test map”, Acta Mathematica Sinica, Chinese Series., 56 (2013), 935-940.
[8]	F. W. Gehring and G. J. Martin., “Commutators, collars and the geometry of Möbius groups”, J. Anal. Math., 63 (1994), 175-219.
[9]	F. W. Gehring and G. J. Martin. “Inequalities for Möbius transformations and discrete groups”, J. Reine Angew. Math., 418 (1991), 31-76.
[10]	F. W. Gehring, J. Gilman and G. J. Martin. “Kleinian groups with real parameters”, Communications in Contemporary Math., 3 (2001), 163-186.
[11]	W. M. Kantor and A. Lubotzky. “The probability ofgroups generating a finite classical group”, Geom. Dedicata., 36 (1990), 67-87.
[12]	M. W. Liebeck and A. Shalev. “The probability ofgenerating generating a finite simple group”, Geom. Dedicata., 56 (1995), 103-113.
[13]	G. J. Martin, G. O’Brien “Random Möbius groups I: Random Subgroups of ”, Pacific. J. Math, 303 (2019), 669-701.
[14]	G. J. Martin, G. O’Brien and Y. Yamashita. “Random Kleinian groups, II: two parabolic generators”, Exp. Math., (2019), 1-9.
[15]	G. J. Martin. “Random ideal hyperbolic quadrilaterals, the cross ratio distribution and punctured tori”, J. London Math. Soc., 2 (2019), 851-870.
[16]	G. J. Martin. “Random Lattices and the Teichmüller metric”, Proc. Amer. Math. Soc., 148 (2019), 289-300.
[17]	B. Masket. “Kleinian Groups”, Springer-Verlag, Berlin., (1987).
[18]	G. Rosenberger. “All generating pairs of all two-generator Fuchsian groups”, Arch. Math. (Basel)., 46 (1986), 198-204.
[19]	M. D. Springer. “The algebra of random variables”, Wiley, New York., (1979),.
[20]	D. Tan. “On two generator discrete groups of Möbius transformations”, Proc. Amer. Math. Soc., 106 (1989), 763-770.
[21]	T. Jørgensen, “On discrete groups of Möbius transformations”, Amer. J. Math., 98 (1976), 739-749.

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Binlin Dai, Zekun Li. (2022). Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators. Applied and Computational Mathematics, 11(1), 31-37. https://doi.org/10.11648/j.acm.20221101.13

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ACS Style

Binlin Dai; Zekun Li. Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators. Appl. Comput. Math. 2022, 11(1), 31-37. doi: 10.11648/j.acm.20221101.13

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AMA Style

Binlin Dai, Zekun Li. Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators. Appl Comput Math. 2022;11(1):31-37. doi: 10.11648/j.acm.20221101.13

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@article{10.11648/j.acm.20221101.13,
  author = {Binlin Dai and Zekun Li},
  title = {Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {1},
  pages = {31-37},
  doi = {10.11648/j.acm.20221101.13},
  url = {https://doi.org/10.11648/j.acm.20221101.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221101.13},
  abstract = {Let  be real Möbius groups. For any 2×2 matrix A in  induces real Möbius transformations g by the formula  where . The collection of all real Möbius transformations for which  takes the values 1 forms a group which can be identified with . We write  to mean that f is a random variable in. In this paper, we study the random Möbius subgroup. We can get some new results: (1) If, the probability  is greater than 0.027282. (2) If f is hyperbolic transformation and g is parabolic transformation, then the probability  is discrete is greater than 7/12. (3) If f is elliptic of order n and g is elliptic of order 2, then the probability  is discrete is greater than 2/n. (4) The probability that random chosen  generate an elementary or non-discrete groupis greater than 0.0302049.},
 year = {2022}
}

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TY  - JOUR
T1  - Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators
AU  - Binlin Dai
AU  - Zekun Li
Y1  - 2022/02/19
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221101.13
DO  - 10.11648/j.acm.20221101.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 31
EP  - 37
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221101.13
AB  - Let  be real Möbius groups. For any 2×2 matrix A in  induces real Möbius transformations g by the formula  where . The collection of all real Möbius transformations for which  takes the values 1 forms a group which can be identified with . We write  to mean that f is a random variable in. In this paper, we study the random Möbius subgroup. We can get some new results: (1) If, the probability  is greater than 0.027282. (2) If f is hyperbolic transformation and g is parabolic transformation, then the probability  is discrete is greater than 7/12. (3) If f is elliptic of order n and g is elliptic of order 2, then the probability  is discrete is greater than 2/n. (4) The probability that random chosen  generate an elementary or non-discrete groupis greater than 0.0302049.
VL  - 11
IS  - 1
ER  -

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Author Information

Binlin Dai

School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China
Zekun Li

School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China

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Binlin Dai, Zekun Li. (2022). Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators. Applied and Computational Mathematics, 11(1), 31-37. https://doi.org/10.11648/j.acm.20221101.13

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ACS Style

Binlin Dai; Zekun Li. Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators. Appl. Comput. Math. 2022, 11(1), 31-37. doi: 10.11648/j.acm.20221101.13

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AMA Style

Binlin Dai, Zekun Li. Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators. Appl Comput Math. 2022;11(1):31-37. doi: 10.11648/j.acm.20221101.13

Copy | Download

@article{10.11648/j.acm.20221101.13,
  author = {Binlin Dai and Zekun Li},
  title = {Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {1},
  pages = {31-37},
  doi = {10.11648/j.acm.20221101.13},
  url = {https://doi.org/10.11648/j.acm.20221101.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221101.13},
  abstract = {Let  be real Möbius groups. For any 2×2 matrix A in  induces real Möbius transformations g by the formula  where . The collection of all real Möbius transformations for which  takes the values 1 forms a group which can be identified with . We write  to mean that f is a random variable in. In this paper, we study the random Möbius subgroup. We can get some new results: (1) If, the probability  is greater than 0.027282. (2) If f is hyperbolic transformation and g is parabolic transformation, then the probability  is discrete is greater than 7/12. (3) If f is elliptic of order n and g is elliptic of order 2, then the probability  is discrete is greater than 2/n. (4) The probability that random chosen  generate an elementary or non-discrete groupis greater than 0.0302049.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Discrete Probability of Random Möbius Groups: Random Subgroups by Two Generators
AU  - Binlin Dai
AU  - Zekun Li
Y1  - 2022/02/19
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221101.13
DO  - 10.11648/j.acm.20221101.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 31
EP  - 37
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221101.13
AB  - Let  be real Möbius groups. For any 2×2 matrix A in  induces real Möbius transformations g by the formula  where . The collection of all real Möbius transformations for which  takes the values 1 forms a group which can be identified with . We write  to mean that f is a random variable in. In this paper, we study the random Möbius subgroup. We can get some new results: (1) If, the probability  is greater than 0.027282. (2) If f is hyperbolic transformation and g is parabolic transformation, then the probability  is discrete is greater than 7/12. (3) If f is elliptic of order n and g is elliptic of order 2, then the probability  is discrete is greater than 2/n. (4) The probability that random chosen  generate an elementary or non-discrete groupis greater than 0.0302049.
VL  - 11
IS  - 1
ER  -

Copy | Download