In a book I'm reading it says: Putting $f(x,y)=F(u,v)$ with $\gamma(x+yi)=u+iv$ and using Cauchy-Riemann equation for $\gamma(z)$, we have
$$\Delta f(x,y) \overset{(*)}=-y^2(u_x^2+v_x^2)(F_{uu}+F_{vv})\overset{(**)}=-y^2|\frac{d}{dz}\gamma(z)|^2 (F_{uu}+F_{vv})=-v^2(F_{uu}+F_{vv}),$$
which amounts to $\Delta \cdot \gamma = \gamma \cdot \Delta$, i.e., the invariance of $\Delta$. Here $$\Delta = -y^2\left(\left(\frac{\partial}{\partial x}\right)^2 + \left(\frac{\partial}{\partial y}\right)^2\right)$$ is the noneuclidean laplacian. And $\gamma$ is an element of modular group $\Gamma$ (I think).
I can't really see why we have the equalities (*) and (**). And why does this tell us the invariance (can't see that either)?
Equality $(*)$ is simply the chain rule applied to $f=F\circ \gamma$. Equation $(**)$ comes from $$ \left|\frac{d\gamma}{dz}\right|^2=\frac{d\gamma}{dz}\frac{d\gamma(\bar z)}{d\bar z} $$