Combinatorics of double cosets and fundamental domains for the subgroups of the modular group

Details Combinatorics of double cosets and fundamental domains for the subgroups of the modular group Combinatorics of double cosets and fundamental domains for the subgroups of the modular group

2009-01-10

arxiv.org/abs/0901.1340v1

Mathematics

Number Theory

12 pages, LateX2e

Scientific paper

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 $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the right cosets $G\setminus\mathrm{PSL}_2(\mathbb{Z})$. 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 $\Gamma_0(N)$, $\Gamma_1(N)$, $\Gamma(N)$ etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in $\log N$. We also give algorithms to locate a given element of the upper half-plane on the fundamental domain and to write a given element of $G$ as a product of independent generators.

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