An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. distances and angles).
The isometries of the hyperbolic plane form a group under composition.
An isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. We first discuss orientation-preserving isometries.
Orientation-preserving isometries
The orientation-preserving isometries of the hyperbolic plane form a subgroup. This subgroup equals \(PSL(2, \mathbf{R})\) (e.g. by the latter group's action on the upper half-plane model of the hyperbolic plane). We will sometimes discuss isometries from the perspective of \(PSL(2, \mathbf{R})\), so for the purposes of this section, we'll assume we have an identification of the group of orientation-preserving isometries with \(PSL(2, \mathbf{R})\).
Thus, given an orientation-preserving isometry of the plane, we can view it as an element of \(PSL(2, \mathbf{R})\). We can then choose a matrix \(M\) in \(SL(2, \mathbf{R})\) representing that element. This matrix will be unique up to multiplication by \(-1\). We will call such an \(M\) a matrix representing the isometry.
There are three types of orientation-preserving isometries of the hyperbolic plane: hyperbolic, elliptic, and parabolic. (This terminology can be confusing at first because "hyperbolic isometry" can be used both to mean any isometry of the hyperbolic plane, as well as only those isometries that have hyperbolic type. In general, readers will have to pay attention to the context to know which meaning is being used.)
There is also a fourth type of isometry sometimes left out because it is trivial: the identity isometry. The identity isometry can also be thought of as a degenerate form of the other three types (e.g. an elliptic isometry with an angle of rotation of zero).
Whether an isometry is hyperbolic, elliptic, parabolic, or the identity is invariant under conjugacy in the full group of isometries of the hyperbolic plane.
Let \(H\) denote the hyperbolic plane, and let \(\overline{H}\) denote the hyperbolic plane together with its boundary at infinity. One easy way to tell the type of an isometry is by the nature and number of its fixed points in \(\overline{H}\). For example, the identity isometry is the unique isometry that fixes all points in \(H\) (as well as in \(\overline{H}\)). We discuss the remaining types below.
Hyperbolic isometry
A hyperbolic isometry is an orientation-preserving isometry with exactly two fixed points in \(\overline{H}\). These two points will necessarily be in the boundary at infinity.
A hyperbolic isometry is sometimes called a "translation." A hyperbolic isometry fixes (as a set) the geodesic that connects its two fixed points. This geodesic is sometimes called its "axis." The isometry translates all points in this geodesic by a common distance. This distance is called the isometry's translation length.
A hyperbolic isometry is determined by (1) its two fixed points, or equivalently the geodesic it fixes, (2) a direction, or orientation of the geodesic it fixes, and (3) its translation length. The translation length is invariant under conjugation in the full group of isometries of the plane, and all hyperbolic isometries with the same translation length are conjugate to one another via an orientation-preserving isometry.
Choose a hyperbolic isometry, and choose a matrix \(M\) in \(SL(2, \mathbf{R})\) representing that isometry. It is a property of hyperbolic isometries that \(|\operatorname{tr}(M)|>2\). Also, the translation length can be computed from the trace as \(2\cosh^{-1}(\operatorname{tr}(M)/2)\).
In addition, the matrix \(M\) is conjugate in \(SL(2, \mathbf{R})\) to a matrix of the form \(\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}\). In particular, \(M\) has two real eigenvalues that are reciprocals. Using this formulation, the translation length can be expressed as \(|\log(\lambda^2)|=2|\log |\lambda||\). This expression can be derived from the expression using the trace by applying the identity \(\cosh^{-1}x=\log(x+\sqrt{x^2 - 1})\) (valid for \(x\geq 1\)).
Elliptic isometry
An elliptic isometry is an isometry with exactly one fixed point in \(\overline{H}\), and with the fixed point in \(H\) rather than in the boundary.
An elliptic isometry is sometimes called a "rotation." An elliptic isometry is determined by (1) its fixed point, and (2) its angle of rotation. An isometry's angle of rotation is invariant under conjugation by orientation-preserving isometries and is invariant up to sign under conjugation in the full group of isometries. All elliptic isometries with the same angle of rotation are conjugate to one another via an orientation-preserving isometry.
Let \(\theta\) be any nontrivial angle of rotation (i.e. so that \(0<\theta<2\pi\)). Consider in the upper half-plane model of the hyperbolic plane the elliptic isometry with fixed point the "center" of the upper half-plane \(i\) and whose angle of rotation is \(\theta\) in the counter-clockwise direction. This isometry is represented by the matrix \(M=\begin{pmatrix} \cos\frac{\theta}{2} & \sin\frac{\theta}{2} \\ -\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{pmatrix}\) in \(SL(2, \mathbf{R})\).
Observe that the trace equals \(\operatorname{tr}(M)=2\cos \frac{\theta}{2}\). Since \(0<\theta<2\pi\), the trace satisfies \(|\operatorname{tr}(M)|<2\). Moreover, the angle of rotation can be recovered from the trace as \(\theta = 2\cos^{-1}(\operatorname{tr}(M)/2)\). This parallels the formula for the translation length in terms of the trace in the hyperbolic case.
The angle of rotation can similarly be recovered from the eigenvalues of \(M\). The eigenvalues of \(M\) can be computed to be the complex numbers \(e^{i\theta/2}\) and \(e^{-i\theta/2}\). If we let \(\lambda=\lambda_1=e^{i\theta/2}\) and \(\lambda_2=e^{-i\theta/2}\) be the eigenvalues, then we have \(|\log(\lambda_1/\lambda_2)|=|\log(\lambda^2)|=\theta\). In other words, \(\theta=2|\log(\lambda)|\). Just as for the trace, this parallels the corresponding expression in the hyperbolic isometry case.
Since any elliptic isometry is conjugate to one that fixes \(i\) in the upper half-plane, and because the eigenvalues and trace are invariant under conjugation, the formulas above hold for any matrix \(M\) representing an elliptic isometry. In particular, \(|\operatorname{tr}(M)|<2\), and, letting \(\lambda\) be an eigenvalue, for the angle of rotation we have \(\theta=2|\log(\lambda)|=2\cos^{-1}(\operatorname{tr}(M)/2)\).
Parabolic isometry
A parabolic isometry is an isometry with exactly one fixed point in \(\overline{H}\), and with the fixed point in the boundary rather than in \(H\).
Choose a point \(p\) in \(H\) (so not on the boundary). A parabolic isometry is then determined by (1) its fixed point, (2) the distance it moves the point \(p\), and (3) the direction it moves \(p\) (e.g. left or right when facing \(H\) from the fixed point). The direction of a parabolic isometry is invariant under conjugation by orientation-preserving isometries but not by orientation-reversing isometries. In the full group of isometries of the plane, all parabolic isometries are conjugate to one another (i.e. there is just one conjugacy class). In the group of orientation-preserving isometries, all parabolic isometries having the same direction are conjugate to one another (so there are two conjugacy classes in this case).
Choose a parabolic isometry, and choose a matrix \(M\) in \(SL(2, \mathbf{R})\) representing that isometry. It is a property of parabolic isometries (and of the identity isometry) that \(|\operatorname{tr}(M)|=2\).
Orientation-reversing isometries
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