2 minor edited Jun 6 '16 at 22:02 doetoe 4,58322 gold badges1818 silver badges3737 bronze badges This is not really an answer (the answer is ACuriousMind's comment: this is a double coset space), but it may help to consider the construction of the moduli space of elliptic curves, as this can be done in the same way but is very easy. Every complex elliptic curve is obtained as $$\Bbb C$$ modulo a lattice. Scaling the lattice by a complex number gives an isomorphic curve, so you can scale in such a way that the lattice is generated by 1 and by $$\tau\in\Bbb H$$, the complex upper half plane (this is similar to gauge fixing). Not all $$\tau$$ give different lattices: two sets of generators give the same lattice if they are related by an element of $$GL_2(\Bbb Z)$$ (there is some residual gauge freedom). Since we work with oriented bases, we can restrict to $$SL_2(\Bbb Z)$$. It is not hard to see that the action of $$SL_2(\Bbb Z)$$ on a basis $$1,\tau$$ corresponds to an action on $$\Bbb H$$ by Möbius transformations $$\begin{pmatrix}a & b\\ c & d\end{pmatrix}\tau = \frac{a\tau + b}{c\tau + d}$$ This gives us the moduli space as a quotient $$\mathcal M \cong SL_2(\Bbb Z)\backslash \Bbb H$$ In general, if you have a space (possibly with some extra structure like a Riemannian metric) on which some group of automorphisms acts transitively (i.e. every point can be mapped to every point) by some mapping of the whole space onto itself, then this space can be written as the quotient of this group by the stabilizer (i.e. the subgroup fixing a given point) of any point. This is the orbit-stabilizer theorem. In our example, the complex upper half plane $$\Bbb H$$ has an obvious complex structure as a subset of $$\Bbb C$$, and its group of holomorphic automorphisms is $$SL_2(\Bbb R)$$ acting by Möbius transformations, except that $$+I$$ and $$-I$$ do the same thing, and the automorphisms are really $$SL_2(\Bbb R)/\langle-I\rangle = PSL_2(\Bbb R)$$. The stabilizer of the point $$i$$ is $$SO_2(\Bbb R)\subset PSL_2(\Bbb R)$$, so that $$\mathcal M \cong SL_2(\Bbb Z)\backslash PSL_2(\Bbb R)/SO_2(\Bbb R)$$ This is not really an answer (the answer is ACuriousMind's comment: this is a double coset space), but it may help to consider the construction of the moduli space of elliptic curves. Every complex elliptic curve is obtained as $$\Bbb C$$ modulo a lattice. Scaling the lattice by a complex number gives an isomorphic curve, so you can scale in such a way that the lattice is generated by 1 and by $$\tau\in\Bbb H$$, the complex upper half plane (this is similar to gauge fixing). Not all $$\tau$$ give different lattices: two sets of generators give the same lattice if they are related by an element of $$GL_2(\Bbb Z)$$ (there is some residual gauge freedom). Since we work with oriented bases, we can restrict to $$SL_2(\Bbb Z)$$. It is not hard to see that the action of $$SL_2(\Bbb Z)$$ on a basis $$1,\tau$$ corresponds to an action on $$\Bbb H$$ by Möbius transformations $$\begin{pmatrix}a & b\\ c & d\end{pmatrix}\tau = \frac{a\tau + b}{c\tau + d}$$ This gives us the moduli space as a quotient $$\mathcal M \cong SL_2(\Bbb Z)\backslash \Bbb H$$ In general, if you have a space (possibly with some extra structure like a Riemannian metric) on which some group of automorphisms acts transitively (i.e. every point can be mapped to every point) by some mapping of the whole space onto itself, then this space can be written as the quotient of this group by the stabilizer (i.e. the subgroup fixing a given point) of any point. This is the orbit-stabilizer theorem. In our example, the complex upper half plane $$\Bbb H$$ has an obvious complex structure as a subset of $$\Bbb C$$, and its group of holomorphic automorphisms is $$SL_2(\Bbb R)$$ acting by Möbius transformations, except that $$+I$$ and $$-I$$ do the same thing, and the automorphisms are really $$SL_2(\Bbb R)/\langle-I\rangle = PSL_2(\Bbb R)$$. The stabilizer of the point $$i$$ is $$SO_2(\Bbb R)\subset PSL_2(\Bbb R)$$, so that $$\mathcal M \cong SL_2(\Bbb Z)\backslash PSL_2(\Bbb R)/SO_2(\Bbb R)$$ This is not really an answer (the answer is ACuriousMind's comment: this is a double coset space), but it may help to consider the construction of the moduli space of elliptic curves, as this can be done in the same way but is very easy. Every complex elliptic curve is obtained as $$\Bbb C$$ modulo a lattice. Scaling the lattice by a complex number gives an isomorphic curve, so you can scale in such a way that the lattice is generated by 1 and by $$\tau\in\Bbb H$$, the complex upper half plane (this is similar to gauge fixing). Not all $$\tau$$ give different lattices: two sets of generators give the same lattice if they are related by an element of $$GL_2(\Bbb Z)$$ (there is some residual gauge freedom). Since we work with oriented bases, we can restrict to $$SL_2(\Bbb Z)$$. It is not hard to see that the action of $$SL_2(\Bbb Z)$$ on a basis $$1,\tau$$ corresponds to an action on $$\Bbb H$$ by Möbius transformations $$\begin{pmatrix}a & b\\ c & d\end{pmatrix}\tau = \frac{a\tau + b}{c\tau + d}$$ This gives us the moduli space as a quotient $$\mathcal M \cong SL_2(\Bbb Z)\backslash \Bbb H$$ In general, if you have a space (possibly with some extra structure like a Riemannian metric) on which some group of automorphisms acts transitively (i.e. every point can be mapped to every point) by some mapping of the whole space onto itself, then this space can be written as the quotient of this group by the stabilizer (i.e. the subgroup fixing a given point) of any point. This is the orbit-stabilizer theorem. In our example, the complex upper half plane $$\Bbb H$$ has an obvious complex structure as a subset of $$\Bbb C$$, and its group of holomorphic automorphisms is $$SL_2(\Bbb R)$$ acting by Möbius transformations, except that $$+I$$ and $$-I$$ do the same thing, and the automorphisms are really $$SL_2(\Bbb R)/\langle-I\rangle = PSL_2(\Bbb R)$$. The stabilizer of the point $$i$$ is $$SO_2(\Bbb R)\subset PSL_2(\Bbb R)$$, so that $$\mathcal M \cong SL_2(\Bbb Z)\backslash PSL_2(\Bbb R)/SO_2(\Bbb R)$$ 1 answered Jun 6 '16 at 21:56 doetoe 4,58322 gold badges1818 silver badges3737 bronze badges This is not really an answer (the answer is ACuriousMind's comment: this is a double coset space), but it may help to consider the construction of the moduli space of elliptic curves. Every complex elliptic curve is obtained as $$\Bbb C$$ modulo a lattice. Scaling the lattice by a complex number gives an isomorphic curve, so you can scale in such a way that the lattice is generated by 1 and by $$\tau\in\Bbb H$$, the complex upper half plane (this is similar to gauge fixing). Not all $$\tau$$ give different lattices: two sets of generators give the same lattice if they are related by an element of $$GL_2(\Bbb Z)$$ (there is some residual gauge freedom). Since we work with oriented bases, we can restrict to $$SL_2(\Bbb Z)$$. It is not hard to see that the action of $$SL_2(\Bbb Z)$$ on a basis $$1,\tau$$ corresponds to an action on $$\Bbb H$$ by Möbius transformations $$\begin{pmatrix}a & b\\ c & d\end{pmatrix}\tau = \frac{a\tau + b}{c\tau + d}$$ This gives us the moduli space as a quotient $$\mathcal M \cong SL_2(\Bbb Z)\backslash \Bbb H$$ In general, if you have a space (possibly with some extra structure like a Riemannian metric) on which some group of automorphisms acts transitively (i.e. every point can be mapped to every point) by some mapping of the whole space onto itself, then this space can be written as the quotient of this group by the stabilizer (i.e. the subgroup fixing a given point) of any point. This is the orbit-stabilizer theorem. In our example, the complex upper half plane $$\Bbb H$$ has an obvious complex structure as a subset of $$\Bbb C$$, and its group of holomorphic automorphisms is $$SL_2(\Bbb R)$$ acting by Möbius transformations, except that $$+I$$ and $$-I$$ do the same thing, and the automorphisms are really $$SL_2(\Bbb R)/\langle-I\rangle = PSL_2(\Bbb R)$$. The stabilizer of the point $$i$$ is $$SO_2(\Bbb R)\subset PSL_2(\Bbb R)$$, so that $$\mathcal M \cong SL_2(\Bbb Z)\backslash PSL_2(\Bbb R)/SO_2(\Bbb R)$$