Fuchsian group

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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 thus is a lattice in one of the two semisimple Lie groups PSL(2,R) or PSL(2,C). They are named in honour of Immanuel Lazarus Fuchs.

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).

1 Fuchsian groups on the upper half-plane
2 General Definition
3 Limit sets
4 Examples
5 See also
6 References

[edit] Fuchsian groups on the upper half-plane

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

$ds=\frac{\sqrt{dx^2+dy^2}}{y}$ .

The group PSL(2,R) acts on H by linear fractional transformations:

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

This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientation-preserving isometries of H.

A Fuchsian group Γ may be defined to be a subgroup of PSL(2,R), which acts discontinuously on H. That is,

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

An equivalent definition for Γ to be Fuchsian is that Γ be discrete, in the following sense:

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.

These properties are exactly what is required to make the quotient space H/Γ a Hausdorff topological space.

Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the Riemann sphere. Indeed, the Fuchsian group PSL(2,Z) is discrete but has accumulation points on the real number line Im z = 0; elements of PSL(2,Z) will carry z = 0 to every rational number; the rationals Q are dense in R.

[edit] General Definition

A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere $\hat{\mathbb {C}}=\mathbb{C}\cup\infty$ , but will send the upper-half plane H to some open disk Δ. Such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ.

This motivates the following definition of a Fuchsian group. Let $\Gamma \subset PSL(2,\mathbb{C})$ act invariantly on a proper, open disk $\Delta\subset \hat{\mathbb {C}}$ , that is, $Γ(Δ) = Δ$ . Then Γ is Fuchsian if and only if any of the following three properties hold:

Γ is a discrete group (with respect to the standard topology on PSL(2,C)).
Γ acts properly discontinuously at each point $z\in\Delta$ .
The set Δ is a subset of the region of discontinuity Ω(Γ) of Γ.

That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picard group PSL(2,Z[i]) is discrete but does not preserve any disk in the Riemann sphere. Indeed, even the modular group PSL(2,Z), which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational numbers. Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.

It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.

[edit] Limit sets

Because of the discrete action, the orbit $Γ z$ of a point z in the upper half-plane under the action of Γ has no accumulation points in the upper half-plane. There may, however, be limit points on the real axis. Let $Λ(Γ)$ be the limit set of $Γ$ , that is, the set of limit points of $Γ z$ for $z \in \mathbb{H}$ . Then $\Lambda(\Gamma) \subseteq \mathbb{R} \cup \infty$ . The limit set may be empty, or may contain one or two points, or may contain an infinite number. In the later case, there are two types:

A Fuchsian group of the first kind is a group for which the limit set is the closed real line $\mathbb{R} \cup \infty$ . This happens if the quotient space H/Γ has finite volume.

Otherwise, a Fuchsian group is said to be of the second kind. Equivalently, this is a group for which the limit set is a perfect set that is nowhere dense on $\mathbb{R} \cup \infty$ . Since it is nowhere dense, this implies that any limit point is arbitrarily close to an open set that is not in the limit set. In other words, the limit set is a Cantor set.

[edit] Examples

By far the most prominent example of a Fuchsian group is the modular group, PSL(2,Z). This is the subgroup of PSL(2,R) consisting of linear fractional transformations

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

where a,b,c,d are integers. The quotient space H/PSL(2,Z) is the moduli space of elliptic curves.

Other famous Fuchsian groups include the groups $Γ$ (n) for each integer n > 0. Here $Γ$ (n) consists of linear fractional transformations of the above form where the entries of the matrix

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

are congruent to those of the identity matrix modulo n.

All these are Fuchsian groups of the first kind.

All hyperbolic and parabolic cyclic subgroups of PSL(2,R) are Fuchsian.

Any elliptic cyclic subgroup is Fuchsian if and only if it is finite.

Every abelian Fuchsian group is cyclic.

No Fuchsian group is isomorphic to $\mathbb{Z} \times \mathbb{Z}$ .

Let $Γ$ be a non-abelian Fuchsian group. Then the normalizer of $Γ$ in PSL(2,R) is Fuchsian.

[edit] See also

[edit] References

David Mumford, Caroline Series and David Wright, Indra's Pearls; The Vision of Felix Klein, (2002) Cambridge University Press ISBN 0-521-35253-3. (Aimed at non-mathematicians, provides an excellent exposition of theory and results, richly illustrated with diagrams.)
Svetlana Katok, Fuchsian Groups (1992), University of Chicago Press, Chicago ISBN 0-226-42583-5
Hershel M. Farkas, Irwin Kra, Theta Constants, Riemann Surfaces and the Modular Group, American Mathematical Society, Providence RI, ISBN 0-8218-1392-7 (See section 1.6)
Peter J. Nicholls, The Ergodic Theory of Discrete Groups, (1989) London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge ISBN 0-521-37674-2
Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), America Mathematical Society, Providence, RI ISBN 0-8218-3160-7 (See Chapter 2).