Finite-index modular groups as cofinite Fuchsian groups #
Every finite-index subgroup of SL(2, ℤ) gives a cofinite subgroup of PSL(2, ℝ).
Its parabolic fixed points are exactly the rational boundary points: equivalently, the
orbit of infinity under the full modular group. The subgroup can split this one modular
orbit into several cusp orbits: every cusp orbit is that of γ • ∞ for some γ ∈ SL(2, ℤ),
and two such orbits agree exactly when γ⁻¹ h γ' fixes ∞ for some h in the subgroup.
These facts allow the compactified Fuchsian quotient and its cusp charts to be used for
congruence modular curves, and reduce counting cusps to arithmetic in SL(2, ℤ); in
particular there are finitely many cusp orbits.
References #
- Fred Diamond and Jerry Shurman, A First Course in Modular Forms, §§2.3–2.4.
A finite-index subgroup of SL(2, ℤ) has cofinite effective image in PSL(2, ℝ).
This formulation applies directly to Γ(N), Γ₀(N), and Γ₁(N).
The parabolic fixed points of a finite-index integral modular subgroup are the rational boundary points, described as the full modular orbit of infinity.
The cusp points of a finite-index subgroup of SL(2, ℤ) are exactly the image of
ℙ¹(ℚ) in the real projective boundary.
The point γ • ∞ of the projective boundary is a cusp point of the effective image of a
finite-index subgroup of SL(2, ℤ).
The cusp orbit of γ • ∞ for the effective image of a finite-index subgroup H of
SL(2, ℤ).
Instances For
The cusp orbit of γ • ∞, as a boundary orbit.
Every cusp orbit of a finite-index subgroup of SL(2, ℤ) is the orbit of γ • ∞ for some
γ ∈ SL(2, ℤ).
The cusp orbits of γ • ∞ and γ' • ∞ agree exactly when γ⁻¹ h γ' fixes ∞, that is
has vanishing lower-left entry, for some h ∈ H.
The matrices γ and -γ represent the same cusp orbit.
A finite-index subgroup of SL(2, ℤ) has finitely many cusp orbits, since its image in
PSL(2, ℝ) is cofinite.