Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Modular group of torus

  1. Feb 25, 2012 #1
    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.
  2. jcsd
  3. Feb 26, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    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).
    Last edited: Feb 26, 2012
  4. Feb 26, 2012 #3
    Cool. That makes sense. Actually, I just read that GL(2,Z) is called the "extended modular group".
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook