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Modular group of torus

by electroweak
Tags: dehn tiwst, isotopy, modular group, torus
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Feb25-12, 03:33 PM
P: 44
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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Feb26-12, 11:41 AM
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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).
Feb26-12, 09:17 PM
P: 44
Cool. That makes sense. Actually, I just read that GL(2,Z) is called the "extended modular group".

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