2009/01/10 by Gorinov, Alexey G., Kalinkin, Isaac C.
#FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0901.1340
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 G\setminusPSL2(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 Γ0(N), Γ1(N), Γ(N) etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in log 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\.