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In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. The model is commonly expressed in terms of the complex upper half plane or the unit disc, both of which are related through a conformal mapping.

The Poincaré metric is a metric tensor on the disk or plane, expressing the hyperbolic nature.

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Symmetry groups

A variety of different groups appear in the discussion of the upper half plane. One is the linear group GL(2,C), called the Möbius group. The projective linear group PGL(2,C) is isomorphic to the group of all orientation-preserving conformal maps of the upper half plane. Important subgroups of PGL(2,C) are called Kleinian groups. However, conformal maps do not in general preserve the Poincaré metric and thus are not an isometry of the Poincaré model.

There are four groups that do preserve the metric tensor. These are:

  • The special linear group SL(2,R) which consists of the set of 2x2 matrices with real entries whose determinant equals +1. Note that many texts (including Wikipedia) often say SL(2,R) when they really mean PSL(2,R).
  • The group S*L(2,R) consisting of the set of 2x2 matrices with real entries whose determinant equals +1 or -1. Note that SL(2,R) is a subgroup of this group.
  • The projective linear group PSL(2,R)=SL(2,R)/{+ or - I}, consisting of the matrices in SL(2,R) modulo plus or minus the identity matrix.
  • The group PS*L(2,R)=S*L(2,R)/{+ or - I} is again a projective group, and again, modulo plus or minus the identity matrix.

The relationship of these groups to the Poincaré model is as follows:

  • The group of all isometries of H, sometimes denoted as Isom(H), is isomorphic to PS*L(2,R). This includes both the orientation preserving and the orientation-reversing isometries. The orientation-reversing map (the mirror map) is <math>z\rightarrow -\overline{z}<math>.
  • The group of orientation-preserving isometries of H, sometimes denoted as Isom+(H), is isomorphic to PSL(2,R).

Important subgroups of the isometry group are the Fuchsian groups.

One also frequently sees the modular group SL(2,Z). This group is important in two ways. First, it is a symmetry group of the square 2x2 lattice of points. Thus, functions that are periodic on a square grid, such as modular forms and elliptic functions, will thus inherit an SL(2,Z) symmetry from the grid. Second, SL(2,Z) is of course a subgroup of SL(2,R), and thus has a hyperbolic behavior embedded in it. In particular, SL(2,Z) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area.

Isometric symmetry

The group action of the special linear group PSL(2,R) on H is defined by

<math>\left(\begin{matrix}a&b\\ c&d\\ \end{matrix}\right) \cdot z = \frac{az+b}{cz+d} = {(ac|z|^2+bd+(ad+bc)\Re(z)) + i\Im(z)\over|cz+d|^2}.<math>

Note that the action is transitive, in that for any <math>z_1,z_2\in\mathbb{H}<math>, there exists a <math>g\in {\rm PSL}(2,\mathbb{R})<math> such that <math>gz_1=z_2<math>. It is also faithful, in that if <math>gz=z<math> for all z in H, then g=e.

The stabilizer or isotropy subroup of an element z in H is the set of <math>g\in{\rm PSL}(2,\mathbb{R})<math> leave z unchanged: gz=z. The stabilizer of i is the rotation group

<math>{\rm SO}(2) = \left\{ \left(\begin{matrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{matrix}\right)\,:\,\theta\in{\mathbf R}\right\}.<math>

Since any element z in H is mapped to i by an element of PSL(2,R), this means that the isotropy subgroup of any z is isomorphic to SO(2). Thus, H = PSL(2,R)/SO(2). The upper half plane is tessellated into free regular sets by the modular group SL(2,Z).

Geodesics

The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis.

See also

References

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