Combinatorics of double cosets and fundamental domains for the subgroups of the modular group
Abstract
As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup G of PSL2(Z) from the combinatorics of the right action of PSL2(Z) on the right cosets G2(Z). This gives a method of constructing nice fundamental domains (which Kulkarni calls "special polygons") for the action of G on the upper half plane. For the classical congruence subgroups 0(N), 1(N), (N) etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in N. This is roughly the square root of the number of operations required by the naive procedure. We give algorithms to locate an element of the upper half-plane on the fundamental domain and to write a given element of G as a product of independent generators. We also (re)prove a few related results about the automorphism groups of modular curves. For example, we give a simple proof that the automorphism group of X(N) is SL2(Z/N)/\ I\.