Poincare half-plane model

The upper half-plane carries the hyperbolic metric $(d{x}^2+d{y}^2)/y^2$.

For a hyperbolic Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$, the infimum of $d(z,Az)$ is attained on its axis and equals $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. It is the length of the corresponding closed geodesic in a quotient.

For a determinant-one lift of a hyperbolic holonomy representation, the squared matrix trace of each fixed group word is a polynomial in the finitely many generator-matrix entries. The displayed identity identifies equal positive lengths with equal squared traces. Lift signs have no effect. Finite choices of length matchings therefore give finite unions of affine algebraic sets.

For $z,w$ in the upper half-plane,

The geodesics in the upper half-plane are vertical lines and semicircles centred on the boundary axis. Indeed, the length integrand $\sqrt{1+y^{\prime 2}}/y$ has the Beltrami identitywhich integrates to $(x-c{)}^2+y^2=R^2$. Swapping the coordinate axes gives circles centred on the vertical boundary for the metric $(d{x}^2+d{y}^2)/x^2$.

Two disjoint hyperbolic lines with four distinct ideal endpoints have a unique common perpendicular, and the segment of that perpendicular between them realizes their minimum distance. An upper-half-plane isometry can make the two lines concentric semicircles; their common perpendicular then becomes the imaginary axis.

Three pairwise ultraparallel hyperbolic lines may have disjoint common perpendiculars or coincident ones. In the upper-half-plane model, three semicircles of radius $1/2$ centred at $-5/2,0,5/2$ have mutually disjoint common perpendiculars. Three concentric semicircles instead have the same vertical common perpendicular. Thus pairwise ultraparallelism alone does not determine the incidence of their perpendiculars.

The hyperbolic metric in the half-plane model is conformal, so hyperbolic and Euclidean angles agree. The line meeting the imaginary axis at $iy$, $y>0$, at signed angle $\alpha \in (0,\pi )$ is represented by the Euclidean circle with centre $y\cot \alpha$ on the real axis and radius $y\csc \alpha$.

PSL2(R) is $S{L}_2(\mathbb{R})/\{\pm I\}$. It acts on the upper half-plane by the Möbius transformationsand is the group of orientation-preserving hyperbolic isometries.

A real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|>2$. It has two fixed ideal boundary points and translates along the geodesic joining them; its hyperbolic translation length is $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. Conjugating by a real Möbius transformation puts it in the form $z\mapsto e^{\ell }z$ with $\ell >0$; its axis is then the imaginary axis.

A real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|<2$. It fixes a point of the Poincare half-plane model; an integral matrix of this type has trace $0$ or $\pm 1$. In the Poincare disc model, moving its fixed point to zero makes it a rotation about zero.

A nonidentity real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|=2$. It has one ideal boundary fixed point and is conjugate to a nonzero translation on the Poincare half-plane model.

Every orientation-preserving isometry of the Poincare half-plane model lies in $PS{L}_2(\mathbb{R})$. Every orientation-reversing isometry is the composition of one of these with the hyperbolic reflection $z\mapsto -\overline{z}$ in the imaginary axis. Equivalently, the full isometry group consists of real Möbius and anti-Möbius transformations, modulo nonzero scalar matrices.

The group $PS{L}_2(\mathbb{R})$ acts transitively on pairs $(\ell ,P)$ consisting of a hyperbolic line and a point on it. Send the two ideal endpoints of $\ell$ to $0$ and $\infty$, then use a positive dilation to move the resulting point on the imaginary axis to $i$.

Reflection in a hyperbolic line is the unique orientation-reversing hyperbolic isometry fixing that line pointwise. The product of reflections in intersecting lines is a rotation through twice their angle.

A hyperbolic half-turn is the unique orientation-preserving isometry of order two fixing a given point. It exchanges every pair of points lying at equal opposite distances along a geodesic through its centre.