Combinatorics of double cosets and fundamental domains for the subgroups of the modular group
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2009
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| _version_ | 1866913832930115584 |
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| author | Gorinov, Alexey G. Kalinkin, Isaac C. |
| author_facet | Gorinov, Alexey G. Kalinkin, Isaac C. |
| contents | 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 $Γ_0(N)$, $Γ_1(N)$, $Γ(N)$ 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 $\mathrm{SL}_2(\mathbb{Z}/N)/\{\pm I\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0901_1340 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | Combinatorics of double cosets and fundamental domains for the subgroups of the modular group Gorinov, Alexey G. Kalinkin, Isaac C. Number Theory Geometric Topology 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 $Γ_0(N)$, $Γ_1(N)$, $Γ(N)$ 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 $\mathrm{SL}_2(\mathbb{Z}/N)/\{\pm I\}$. |
| title | Combinatorics of double cosets and fundamental domains for the subgroups of the modular group |
| topic | Number Theory Geometric Topology |
| url | https://arxiv.org/abs/0901.1340 |