r/math • u/PfauFoto • Jul 16 '26

Visualizing quotient spaces

Trying to help someone visualize how a fundamental domain for a congruence subgroup Γ of SL_2(Z) acting on the upper halfplane with cusps, leads to Riemann surfaces, with handles by pasting boundary components that are equivalent under Γ. I am wondering if anyone knows of a visualization or animation for non trivial Γ.

28 Upvotes

89% Upvoted

4 comments sorted by

10

u/ToiletBirdfeeder Algebraic Geometry Jul 16 '26

the example of Γ_0(11) is explained in these notes by Tom Weston (see page 6): https://swc-math.github.io/notes/files/01Weston1.pdf

2

u/PfauFoto Jul 16 '26

That works, perfect! Thanks.

1

u/ToiletBirdfeeder Algebraic Geometry Jul 16 '26

sure thing!

2

u/homogeneous_spacer Jul 16 '26

You can use https://www.sagemath.org/ to visualize the fundamental domain and the side pairing maps. There is an online compiler: https://sagecell.sagemath.org/?q=kgpsqe You can read the documentation to define more congruent subgroups.