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Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that:

a. The measure $\mu$ is invariant under all $g \in SL_2(\mathbb R)$ i.e. $\mu(gA) = \mu(A)$.
b. Compute the hyperbolic area of the standard fundamental domain for the modular group $SL_2(\mathbb Z)$ ( $ A = \{\tau | -0.5 \leq \Re(\tau) \leq 0.5 , |\tau| \geq 1\} $ ).

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    Yes, you are right :) Yet I think that proving isometry is slightly simpler, and for $PSL(2, \mathbb{R})$ the proof is very beautiful when you identify its elements with a subgroup of Möbius transformations.2011-04-02

1 Answers 1

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Hints:

  1. That the measure is (left) invariant is essentially shown here.

  2. You want to compute the integral $\mu(A) = \int_{-\frac{1}{2}}^{\frac{1}{2}} \int_{\sqrt{1-x^2}}^{\infty} \frac{1}{y^2}\,dy\,dx$ and that shouldn't be too hard. To check your computations, the answer is $\frac{\pi}{3}$.


On Pavel's request I elaborate on point 1.

Write $z = x + iy$ and $w = g\cdot z = \frac{az + b}{cz + d} = u + iv$ where $g = \begin{bmatrix}a & b \\\ c & d \end{bmatrix}\in \operatorname{SL}(2,\mathbb{R})$. Then the Jacobian determinant is $\frac{\partial (u,v)}{\partial(x,y)} = \frac{\partial u}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial u}{\partial y} \frac{\partial v}{\partial x}$ and by appealing to the Cauchy-Riemann equations we see that this is equal to $\frac{\partial (u,v)}{\partial(x,y)} = \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial x}\right)^2 = \cdots = \frac{1}{|cz + d|^4}$ using the calculations in the Wikipedia link I gave above.

Recalling that $v = \frac{y}{|cz + d|^2}$ we can compute $ \mu(gA) = \int\!\!\!\!\int_{gA} \frac{du\,dv}{v^2} = \int\!\!\!\!\int_{A} \frac{|cz + d|^4}{y^2}\,\frac{\partial (u,v)}{\partial(x,y)}\, dx\, dy = \int\!\!\!\!\int_{A} \frac{dx\, dy}{y^2} = \mu(A)$ as we wanted.

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    Thanks. About 1 - I didn't learn tensors, so I didn't understand why it is invariant and how the volume element derived $f$rom there. Can you explain some more?2011-04-02