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Modular group of torus 
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#1
Feb2512, 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 orientationpreservation, but I haven't been able to flesh this out. Thanks in advance.



#2
Feb2612, 11:41 AM

Sci Advisor
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P: 2,020

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 (=selfdiffeomorphisms?) of R^2/Z^2 lie in onetoone 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 orientationpreserving automorphisms, then I believe the above comments show why this group is SL(2,Z). 


#3
Feb2612, 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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