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Concept
Fuchsian group
Summary
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R).
Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).
General Fuchsian groups were first studied by , who was motivated by the paper , and therefore named them after Lazarus Fuchs.
Let H = {z in C : Im(z) > 0} be the upper half-plane. Then H is a model of the hyperbolic plane when endowed with the metric
The group PSL(2,R) acts on H by linear fractional transformations (also known as Möbius 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 a discrete group, which means that:
Every sequence {γn} 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, γn = I, where I is the identity matrix.
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Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry.
Discrete group
In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its identity is isolated. A subgroup H of a topological group G is a discrete subgroup if H is discrete when endowed with the subspace topology from G. In other words there is a neighbourhood of the identity in G containing no other element of H.
Modular curve
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane).