Fuchsian group

In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. A Fuchsian group is always a discrete group, and is a special case of a lattice in a semisimple Lie group.

Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry, but the theory is much richer. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).

Definition

Let H = {z in C : Im(z) > 0} be the upper half-plane. Then H is a model for hyperbolic plane geometry, when given the element of arc length

<math>ds=\frac{\sqrt{dx^2+dy^2}}{y}<math>

The group G = PSL(2,R) acts on H by linear fractional transformation:

<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z = \frac{az + b}{cz + d}<math>

This action is faithful, and in fact G is isomorphic to the group of all orientation-preserving isometries of H.

A Fuchsian group Γ is a discontinuous subgroup of G, which means the following:

• For every z in H, the orbit Γz = {γz : γ in Γ} has no accumulation point in H.

In this particular instance, an equivalent condition for Γ to be discontinuous is that Γ be discrete, which means the following:

• Every sequence {γ_n} of elements of Γ converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer N such that for all n > N, γ_n = I, where I is the identity matrix.

A strong warning should be issued that the properties of discontinuity and discreteness are not equivalent in the general case of an arbitrary group of conformal homeomorphisms of the Riemann sphere.

A Fuchsian group is also torsion free, meaning it has no finite abelian subgroups.

Similar in many ways to the Fuchsian group is the Kleinian group, a discrete subgroup of PSL(2,C) that can be used to define Kleinian models of three-dimensional hyperbolic manifolds, in direct analogy to Fuchsian models.

Examples

By far the most prominent example of a Fuchsian group is the modular group.

References

• Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0–226–42583–5