Fuchsian Group - Fuchsian Groups On The Upper Half-plane

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

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

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 group, 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.

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, and the rationals Q are dense in R.