Automorphisms of the Upper Half Plane

In this note we study the group \(\mathrm{Aut}(\mathbb{H})\) of analytic automorpisms of the upper half plane. We review the usual classification of such automorphisms based on their fixed points on the Riemann sphere \(\mathbb{PC}^1\), and investigate certain subgroups occuring frequently in applications. It is well known that \(\mathrm{Aut}(\mathbb{H})\) can be described explicitly via the isomorphisms

\begin{equation*} \mathrm{GL}^+(2;\mathbb{R})/\mathbb{R}^\times \cong \mathrm{SL}(2;\mathbb{R}) / \{\pm 1\} \cong \mathrm{Aut}(\mathbb{H}) \end{equation*}

where the groups \(\mathrm{GL}^+(2;\mathbb{R})\) and \(\mathrm{SL}(2;\mathbb{R})\) act by fractional linear transformations. We make use of these identifications implicitly throughout.

Classification of automorphisms

In this section we derive a classification of the elements of \(\mathrm{Aut}(\mathbb{H})\) based on their fixed points. Let \(g = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \mathrm{GL}^+(2;\mathbb{R})\) by an arbitrary automorphism, and suppose \(z\) is a fixed point of \(g\). Note that

\begin{equation*} g\cdot\infty = \frac{a\infty + b}{c\infty + d} = \frac{a}{c} \end{equation*}

so that \(g\) fixes \(\infty\) if and only if \(c = 0\). We consider the cases \(c \ne 0\) and \(c = 0\) separately.

Case 1: \(c \ne 0\). In this case we can also suppose \(z \ne \infty\). Then we have

\begin{equation*} g\cdot z = \frac{az + b}{cz + d} = z \quad \Longrightarrow \quad cz^2 + (d-a)z - b = 0 \end{equation*}

Since \(c \ne 0\) the quadratic equation gives solutions

\begin{equation*} \frac{-(d-a) \pm \sqrt{(d-a)^2 + 4cb}}{2c} \end{equation*}

The discriminant of the above is given by

\begin{equation*} (d-a)^2 +4cb = \mathrm{tr}(g)^2 - 4 \mathrm{det}(g) \end{equation*}

We can conclude that:

  • \(\mathrm{tr}(g)^2 > 4\mathrm{det}(g)\) implies \(g\) has two distinct fixed points \(\in \mathbb{R}\).
  • \(\mathrm{tr}(g)^2 = 4\mathrm{det}(g)\) implies \(g\) has a single fixed point \(\in \mathbb{R}\).
  • \(\mathrm{tr}(g)^2 < 4\mathrm{det}(g)\) implies \(g\) has two distinct fixed points \(z, \bar{z} \in \mathbb{C} \setminus \mathbb{R}\).

Case 2: \(c = 0\) In this case, \(\infty\) is automatically a fixed point of \(g\). For \(z\ne \infty\) we have

\begin{equation*} g\cdot z = \frac{a}{d} z + \frac{b}{d} \end{equation*}

If \(a = d\) then this is simply the map \(z\mapsto z + b/d\) which clearly has a single fixed point at \(\infty \in \mathbb{PC}^1\) Moreover, note that \(a = d\) implies \(\mathrm{tr}(g)^2 = 4 \mathrm{det}(g)\). If \(a \ne d\) then solving the fixed point equation \(z = g\cdot z\) for \(z\) yields

\begin{equation*} z = \frac{b}{d-a} \end{equation*}

In other words, \(a \ne d\) implies \(g\) has two fixed points, \(b/(d-a)\) and \(\infty\). We also have \(\mathrm{tr}(g)^2 > 4 \mathrm{det}(g)\) in this case.

Inspired by the roles of \(\mathrm{tr}(g)\) and \(\mathrm{det}\) in the above argument, we make the following definitions:

Definition 1: We say that \(g = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \mathrm{Aut}(\mathbb{H})\) is

  • hyperbolic if \(\mathrm{tr}(g)^2 > 4\mathrm{det}(g)\);
  • parabolic if \(\mathrm{tr}(g)^2 = 4\mathrm{det}(g)\);
  • elliptic if \(\mathrm{tr}(g)^2 < 4\mathrm{det}(g)\). ∎

Using this terminology and Combining cases 1 and 2, we arrive at a classification of elements in \(\mathrm{Aut}(\mathbb{H})\) based on their fixed points in \(\mathbb{PC}^1\):

Theorem 1: Let \(g \in \mathrm{Aut}(\mathbb{H})\) be arbitrary. Then

  • \(g\) hyperbolic implies \(g\) has two distinct fixed points \(\in \mathbb{R} \cup \infty\);
  • \(g\) parabolic implies \(g\) has a single fixed point \(\in \mathbb{R} \cup \infty\).
  • \(g\) elliptic implies \(g\) has two distinct fixed points \(z, \bar{z} \in \mathbb{C} \setminus (\mathbb{R} \cup \infty)\). ∎

Stabilizer subgroups

For convenience, we introduce the following notation for certain stabilizer subgroups.

Definition 2: Let \(z \in \mathbb{H} \cup \mathbb{R} \cup {\infty }\). We set

\begin{align*} \mathrm{GL}^+(2;\mathbb{R})_z &:= \text{ stabilizer of } z \text{ in } \mathbb{GL}^+(2;\mathbb{R}) \\ \mathrm{GL}^+(2;\mathbb{R})^{(p)}_z &:= \text{ parabolic elements of } \mathrm{GL}^+(2;\mathbb{R})_z \end{align*}

The group \(\mathrm{GL}^+(2; \mathbb{R})^{(p)}_z\) is called the parabolic stabilizer of \(z\). ∎

The next result is immediate from our discussion on the first section.

Theorem 2: The stabilizer of \(\infty\) in \(\mathrm{GL}^+(2; \mathbb{R})\) is given by

\begin{equation*} \mathrm{GL}^+(2;\mathbb{R})_z = \left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} ~\colon~ a, d \in \mathbb{R},~ ad > 0,~ b \in \mathbb{R}\right\} \end{equation*}

while the parabolic stabilizer is

\begin{equation*} \mathrm{GL}^+(2;\mathbb{R})^{(p)}_z = \left\{ \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} ~\colon~ a \in \mathbb{R} \setminus \{0\},~ b \in \mathbb{R}\right\} \end{equation*}

∎

Theorem 3: The stabilizer of \(i\) is \(\mathbb{R}^\times \times \mathrm{SO}(2;\mathbb{R})\).

Proof: Suppose \(g = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \mathrm{GL}^+(2;\mathbb{R})\) stabilizes \(i \in \mathbb{H}\). We have

\begin{equation*} \frac{ai + b}{ci + d} = i \quad \Leftrightarrow \quad ai + b = -c + di \end{equation*}

So it is necessary that \(a = d\) and \(b = -c\); in other words, \(g = \left(\begin{smallmatrix} a & b \\ -b & a \end{smallmatrix}\right)\). Computing the determinant gives \(\mathrm{det} \left(\begin{smallmatrix} a & b \\ -b & a \end{smallmatrix} \right) = a^2 + b^2 = R\) for some \(R > 0\). The matrices satisfying this equation are precisely those of the form \(\pm \sqrt{R} \cdot \left(\begin{smallmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{smallmatrix} \right)\) with \(\theta \in [0, 2\pi)\). But this set is precisely \(\mathbb{R}^\times \times \mathrm{SO}(2;\mathbb{R})\), so we are done. ∎