Computing the Modular Group of the Torus

In summary, the conversation discusses the computation of the modular group of the torus. It is mentioned that Dehn twists generate the modular group and are automorphisms of isotopy classes. The modular group is identified as Aut(pi1(T^2))=Aut(Z^2)=GL(2,Z), but it is also known as SL(2,Z) due to its orientation-preserving property. This is demonstrated through the relationship between automorphisms of R^2/Z^2 and elements of GL(2,Z), as well as the determinant criterion for orientation preservation. Therefore, the modular group can be referred to as SL(2,Z) in the context of orientation-preservation.
  • #1
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How does one compute the modular group of the torus? I see how Dehn twists generate the modular group, and I see how Dehn twists are really automorphisms of isotopy classes. Based on this, it seems that the modular group should be Aut(pi1(T^2))=Aut(Z^2)=GL(2,Z). But I've read that the modular group is in fact SL(2,Z). How does this work? I may have something to do with orientation-preservation, but I haven't been able to flesh this out. Thanks in advance.
 
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  • #2
This really isn't my area, but let me give it a shot.

On the one hand, every element in GL(2,Z)=Aut(Z^2) gives us an automorphism of R^2 that stabilizes Z^2 (here I'm using the standard basis for everything), hence descends to an automorphism of the torus R^2/Z^2. On the other hand, every automorphism of R^2/Z^2 induces an automorphism of pi_1(R^2/Z^2) = Z^2 (this equality is really a specific isomorphism). It seems to me everything here is compatible, and that it shouldn't be too hard to conclude that the isotopy classes of automorphisms (=self-diffeomorphisms?) of R^2/Z^2 lie in one-to-one correspondence with elements of GL(2,Z).

The final observation to make is that an automorphism of R^2/Z^2 preserves the orientation defined by the basis {(1,0), (0,1)} for the lattice iff the corresponding automorphism in GL(2,Z) preserves the orientation in R^2 defined by the basis {(1,0),(0,1)} - i.e., iff the corresponding automorphism in GL(2,Z) has positive determinant <=> has determinant 1 (since everything in GL(2,Z) has determinant +/- 1).

So if by "modular group" you mean group of isotopy classes of orientation-preserving automorphisms, then I believe the above comments show why this group is SL(2,Z).
 
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  • #3
Cool. That makes sense. Actually, I just read that GL(2,Z) is called the "extended modular group".
 

What is the modular group of torus?

The modular group of torus is a mathematical concept that represents the symmetry of a torus, which is a doughnut-shaped geometric figure. It consists of all the transformations that preserve the shape and orientation of the torus, including rotations, reflections, and translations.

How is the modular group of torus represented?

The modular group of torus is typically represented by the symbol SL(2,Z), which stands for the special linear group of 2x2 matrices with integer coefficients and determinant 1. This group can also be represented using other matrices, such as the PSL(2,Z) or GL(2,Z) groups.

What is the significance of the modular group of torus?

The modular group of torus has important applications in many areas of mathematics, including number theory, geometry, and topology. It is also closely related to other important mathematical objects, such as the modular curve and the modular form.

How is the modular group of torus related to the concept of modular forms?

The modular group of torus plays a crucial role in the theory of modular forms, which are mathematical functions that satisfy certain transformation properties under the modular group. In fact, these two concepts are intimately connected, as the modular group acts as the symmetry group for modular forms.

What are some real-world examples of the modular group of torus?

The modular group of torus can be applied to various real-world situations, such as the study of crystal structures, the design of computer graphics, and the analysis of DNA sequences. It can also be used to model the behavior of particles in physics and to understand the symmetries of certain chemical compounds.

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