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Modular Group Gamma

The group Gamma of all Möbius transformations of the form

tau^'=(atau+b)/(ctau+d),
(1)

where a, b, c, and d are integers with ad-bc=1. The group can be represented by the 2×2 matrix

A=[a b; c d],
(2)

where det(A)=1. Every A in Gamma can be expressed in the form

A=T^(n_1)ST^(n_2)S...ST^(n_k),
(3)

where

S=[0 -1; 1 0]
(4)
T=[1 1; 0 1],
(5)

although the representation is not unique (Apostol 1997, pp. 28-29).

See also

Klein's Absolute Invariant, Möbius Transformation, Modular Group Gamma0, Modular Group Lambda, Unimodular Transformation

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References

Apostol, T. M. "The Modular Group and Modular Functions." Ch. 2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 17 and 26-46, 1997.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 113, 1987.

Referenced on Wolfram|Alpha

Modular Group Gamma

Cite this as:

Weisstein, Eric W. "Modular Group Gamma." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModularGroupGamma.html

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