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The projective group $\overset{ˉ}{Γ} (2)$ is torsion-free and acts freely

Statement

The only torsion elements of $Γ (2)$ are $\pm I$ ; equivalently $\overset{ˉ}{Γ} (2) = Γ (2) / {\pm I}$ is torsion-free and has no elliptic fixed points on $H$ . In particular $Γ (2)$ has no elliptic points and every point of $H$ has trivial stabiliser in $\overset{ˉ}{Γ} (2)$ .

Facts & Assumptions

Given: $Γ (2) = {γ \in S L_{2} (Z) : γ \equiv I (m o d 2)}$ with image $\overset{ˉ}{Γ} (2) \leq P S L_{2} (Z)$ (The principal congruence subgroup Gamma(2), Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ); the action of $P S L_{2} (Z)$ on $H$ (The modular group and its action on the upper half-plane).

[F1]

A matrix $γ = (\begin{matrix} a & b \\ c & d \end{matrix}) \in Γ (2)$ satisfies $a \equiv d \equiv 1 (m o d 2)$ and $b \equiv c \equiv 0 (m o d 2)$ , and $a d - b c = 1$ (The principal congruence subgroup Gamma(2), Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[F2]

A nonidentity element of $P S L_{2} (R)$ represented by $A \in S L_{2} (R)$ is either parabolic (conjugate to a nonzero translation) or, with two fixed points on $\hat{C}$ , is conjugate to $z \mapsto λ z$ , $λ \neq 0, 1$ , and $tr (A)^{2} = λ + 2 + λ^{- 1}$ ; it is elliptic exactly when $∣ λ ∣ = 1$ (Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the projective trace invariant).

[F3]

Order and torsion in a group; a nonidentity element of $P S L_{2} (R)$ with a fixed point in $H$ is conjugate to a rotation $z \mapsto e^{i θ} z$ , and every point of $H$ with nontrivial stabiliser in $P S L_{2} (Z)$ is $P S L_{2} (Z)$ -equivalent to $i$ , $ω$ or $ω + 1$ (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity, The standard fundamental domain, boundary identifications and elliptic stabilisers, Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the projective trace invariant).

Proof

1.1F1F2givenalgebra

Suppose $γ \in Γ (2)$ has a nontrivial finite-order class in $P S L_{2} (Z)$ . By [F2] it cannot be parabolic, since a nonzero translation has infinite order. Thus it is conjugate to $z \mapsto λ z$ with $λ \neq 1$ a root of unity, and $(tr γ)^{2} = λ + 2 + λ^{- 1} = 2 + 2 \cos θ < 4$ . The trace $a + d$ is an even integer by [F1], so $∣ a + d ∣ < 2$ forces $a + d = 0$ . Hence $d = - a$ , where $a$ is odd, and the determinant equation gives $b c = - a^{2} - 1 \equiv 2 (m o d 4)$ . But $b, c$ are even, so $b c \equiv 0 (m o d 4)$ , a contradiction. Therefore a finite-order projective class is trivial and its representative is $\pm I$ . In particular the only torsion matrices in $Γ (2)$ are $\pm I$ .

2.1F3step 1.1givenalgebra∎

For any $τ \in H$ , its $P S L_{2} (Z)$ -stabiliser is conjugate to a subgroup of the finite stabilisers in the standard domain [F3]. Thus every matrix in $Γ (2)$ fixing $τ$ has a finite-order projective class. By 1.1 it is $\pm I$ , so its class in $\overset{ˉ}{Γ} (2)$ is the identity. Hence $\overset{ˉ}{Γ} (2)$ is torsion-free and acts freely on $H$ , and there are no elliptic points modulo scalars.

Depends on

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Sources

C. T. McMullen, Advanced Complex Analysis, Math 213a course notes (Harvard, 2010) (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)

The projective group Γˉ(2)Γˉ(2) is torsion-free and acts freely