= Solution
The <modular group> is $SL_2(\mathbb Z)$ acting on the <complex upper half-plane> by
$$
\gamma\tau=\frac{a\tau+b}{c\tau+d},\qquad ad-bc=1.
$$
Its central element $-I$ acts trivially on points, so the effective group is $PSL_2(\mathbb Z)$. It is generated by $S:\tau\mapsto-1/\tau$ and $T:\tau\mapsto\tau+1$: the Euclidean algorithm reduces the bottom row of a matrix using these operations until it is a translation. The relations $S^2=(ST)^3=1$ hold in the projective group.
A <standard fundamental domain of the modular group> is
$$
\mathcal F=\{\tau\in\mathbb H:|\operatorname{Re}\tau|\leq1/2,\ |\tau|\geq1\}.
$$
To see that each orbit meets it, use $\operatorname{Im}(\gamma\tau)=\operatorname{Im}\tau/|c\tau+d|^2$. Choose a coprime integer pair $(c,d)$ minimizing the nonzero lattice modulus $|c\tau+d|$; a modular matrix with that bottom row maximizes the imaginary part. Translate to the central strip. If the result had modulus below one, inversion would further increase its imaginary part, a contradiction. Interior representatives are unique: for a point in the interior, $|c\tau+d|>1$ when $c\ne0$. For $|c|\geq2$ use $\operatorname{Im}\tau>\sqrt3/2$; for $|c|=1$ use the strip and the strict unit-circle bound. Thus such a matrix lowers the imaginary part, which would contradict the same fact applied to its inverse if both representatives were in the interior. Matrices with $c=0$ are translations, and cannot move one interior strip point to another.
The vertical sides are identified by $T$, and the circular halves by $S$. The exceptional points with <elliptic stabilizers of the modular group> are represented by $i$ and $\rho=e^{2\pi i/3}$, of projective <stabilizer> orders two and three. The rational boundary points together with infinity form one <modular cusp> orbit. In the hyperbolic metric the region has area $\int_{-1/2}^{1/2}(1-x^2)^{-1/2}dx=\pi/3$.
A useful <subgroup> is the level-two <principal congruence subgroup>,
$$
\Gamma(2)=\{\gamma\in SL_2(\mathbb Z):\gamma\equiv I\pmod2\}.
$$
Reduction modulo two is onto $SL_2(\mathbb F_2)$, which has six elements and permutes the three nonzero vectors. Hence its kernel has index six, and the quotient is $S_3$. In the projective action, $\Gamma(2)$ is torsion-free. Indeed finite-order nonidentity modular elements have trace zero or $\pm1$; matrices in $\Gamma(2)$ have even trace, and trace zero is impossible because $a,d$ are odd while $b,c$ are even: $ad-bc=1$ with $d=-a$ would imply $bc=-a^2-1\equiv2\pmod4$. The <subgroup> has three <modular cusp> classes, represented by $\infty,0,1$, each of width two. Parity of a primitive pair distinguishes these classes, and elementary level-two row operations reduce each class to its representative. Six translates of $\mathcal F$ supply a fundamental region. Its compactified <level-two modular curve> is obtained by adjoining these three <modular cusp> points.
Invariant <holomorphic functions> arise naturally from <theta constants>. In the radian convention of Question 4, put $\Theta_j(\tau)=\theta_j(0,\tau)$, and define
$$
\lambda(\tau)=\frac{\Theta_2(\tau)^4}{\Theta_3(\tau)^4}.
$$
Here $\Theta_2=\sum_{n\in\mathbb Z}q^{(n+1/2)^2}$. The <theta-constant inversion and translation laws>, obtained by Gaussian <Poisson summation> and by reindexing, interchange $\Theta_2,\Theta_4$ under $S$ with the common factor $\sqrt{-i\tau}$, leave $\Theta_3$ fixed up to that factor, and under $T$ interchange $\Theta_3,\Theta_4$ while multiplying $\Theta_2$ by $e^{\pi i/4}$. Together with the <Jacobi abstruse identity> $\Theta_3^4=\Theta_2^4+\Theta_4^4$, this gives
$$
\lambda(S\tau)=1-\lambda(\tau),\qquad
\lambda(T\tau)=\frac{\lambda(\tau)}{\lambda(\tau)-1}.
$$
These transformations generate the six permutations of $0,1,\infty$. In particular $T^2$ and $ST^2S^{-1}$ fix $\lambda$; with $-I$, these generate $\Gamma(2)$. Thus the <modular lambda function> is invariant under this <subgroup>, but not under the whole <modular group>. Its even <theta constants> do not vanish on $\mathbb H$, so it is <holomorphic> there and avoids $0,1,\infty$. Near infinity, $\lambda=16q+O(q^2)$, with $q=e^{\pi i\tau}$, which is the width-two <modular cusp> coordinate. The two transformation laws give <modular cusp> values one at zero and infinity at one, with the same simple local orders. Thus $\lambda$ has exactly one <simple pole> on the compactified level-two quotient and no <pole> in the half-plane. A <meromorphic> map with one <simple pole> has degree one, so it identifies that quotient with the sphere and the open quotient with $\mathbb C\setminus\{0,1\}$. This proves the coordinate assertion and describes all three <modular cusps>.
Symmetrizing this <subgroup> invariant produces a full <modular-invariant function>:
$$
\boxed{j(\tau)=256\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2}}.
$$
Substitution shows that this rational expression is unchanged by both $\lambda\mapsto1-\lambda$ and $\lambda\mapsto\lambda/(\lambda-1)$, hence by $S$ and $T$. It is the <Klein j-invariant>. Its denominator does not vanish on $\mathbb H$, so it is <holomorphic> there; at the <modular cusp> it has $j\sim q^{-2}$, a <simple pole> in $Q=e^{2\pi i\tau}$. It therefore remains nonconstant without being a bounded <holomorphic function> on the compactified quotient. In fact a <holomorphic> invariant extending holomorphically to the compactified <modular curve> would be constant by the <maximum principle>.
This also connects invariance to <elliptic curves>: changing an oriented basis of the lattice $\mathbb Z+\tau\mathbb Z$ gives precisely the modular action after a rescaling. On the compact full modular quotient, $j$ has just the one simple <modular cusp> <pole>, so its map to the sphere has degree one. It therefore classifies the resulting <homothety of complex lattices> classes, while the <modular lambda function> retains a level-two labeling of the <branch points> in the Legendre model $y^2=x(x-1)(x-\lambda)$. Finally, a <modular form> of nonzero weight obeys a transformation law with a factor $(c\tau+d)^k$, not plain invariance; ratios of equal-weight forms instead give weight-zero <modular functions>. The distinction between forms, invariant <holomorphic functions> on the open half-plane, and their <modular cusp> behavior is essential to this construction.
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