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This presentation describes the modular group as the rotational triangle group D(2, 3, ∞) (infinity as there is no relation on T), and it thus maps onto all triangle groups (2, 3, n) by adding the relation T n = 1, which occurs for instance in the congruence subgroup Γ(n).
The modular group is generated by two elements, S and T, subject to the relations S² = (ST)³ = 1 (no relation on T), is the rotational triangle group (2,3,∞) and maps onto all triangle groups (2,3,n) by adding the relation T n = 1.
Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can be visualized by the associated tilings, as depicted at right: the (2,3,7) tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane.
Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain. The diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half-plane H.
It is topologically related to a polyhedra sequence; see discussion.This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the (2,3,n) triangle groups – for the heptagonal tiling, the important (2,3,7) triangle group.
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The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N for some positive integer N, which is defined to be
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