Snippets

# The many faces of the hyperbolic plane

$\mathbb{H}^2$ is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

1. It is diffeomorphic to $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ as a topological space (or, indeed, isometric as a Riemannian manifold, if we give $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ a left-invariant metric): to show this, we note that $\mathbb{H}^2$ has isometry group $\mathrm{SL}(2,\mathbb{R}) / \pm\mathrm{id}$, and the subgroup of isometries which stabilize any given $p \in \mathbb{H}^2$ is isomorphic to $\mathrm{SO}(2)$.
2. Since any positive-definite binary quadratic form is given by a symmetric 2-by-2 matrix with positive eigenvalues, and since the group of linear transformations on $\mathbb{R}^2$ preserving any such given form is isomorphic to $\mathrm{O}(2)$, $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the space of positive-definite binary quadratic forms of determinant 1, via the map from $\mathrm{SL}(2,\mathbb{R})$ to the symmetric positive-definite 2-by-2 matrices given by $g \mapsto g^T g$.
3. $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2$ is also the moduli space of marked Riemann surfaces of genus 1, i.e. the Teichmüller space Teich(S) of the torus S. One way to prove this is to note that any such marked surface is the quotient of $\mathbb{R}^2$ by a $\mathbb{Z}^2$ action; after a suitable conformal transformation, we may assume that the generators of this $\mathbb{Z}^2$ act as $z \mapsto z + 1$ and $z \mapsto z + \tau$ for some $z \in \mathbb{H}^2$ (in the upper half-plane model.) But now $\tau$ is the unique invariant specifying this point in our moduli space.
4. Since any marked Riemann surface of genus 1 has a unique flat metric (inherited as a quotient manifold of the Euclidean plane), $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the moduli space of marked flat 2-tori of unit area.
5. Since there is a unique unit-covolume marked lattice associated to each marked complex torus in the above, $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the space of marked lattices in $\mathbb{R}^2$ with unit covolume.
6. Note we may go directly between marked Riemann surfaces and quadratic forms by considering intersection forms.