Discrete subgroup of the real projective special linear group of dimension 2
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 Henri Poincaré (1882), who was motivated by the paper (Fuchs 1880), and therefore named them after Lazarus Fuchs.
In mathematics, a Fuchsiangroup is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving...
Arithmetic Fuchsiangroups are a special class of Fuchsiangroups constructed using orders in quaternion algebras. They are particular instances of arithmetic...
is called a quasi-Fuchsiangroup. When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsiangroups under conformal transformations...
hyperbolic plane. Fuchsiangroups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A Fuchsiangroup that preserves orientation...
hyperbolic plane). Generalising the example of the modular group a Fuchsiangroup is a group admitting a properly discontinuous action on the hyperbolic...
mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsiangroup. Every hyperbolic...
developments of automorphic forms other than modular forms. The case of Γ a Fuchsiangroup had already received attention before 1900 (see below). The Hilbert...
Riemann surfaces of the same genus using a quasi-Fuchsiangroup of the first kind. The quasi-Fuchsiangroup is essentially uniquely determined by the two...
Henrik Abel in 1827. Bianchi group Classical modular curve Fuchsiangroup J-invariant Kleinian group Mapping class group Minkowski's question-mark function...
field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsiangroups, though related objects...
negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsiangroup. For the following, let H be the...
that can be subsets of Jordan curves Lakes of Wada Quasi-Fuchsiangroup, a mathematical group that preserves a Jordan curve Jordan (1887). Kline, J. R...
listed as a grave of honour of the State of Berlin. He is the eponym of Fuchsiangroups and functions, and the Picard–Fuchs equation. A singular point a of...
isomorphic to a quotient of the upper half-plane by a Fuchsiangroup (this is sometimes called a Fuchsian model for the surface). The topological type of X...
Poincaré and Klein introduced the group of Möbius transformations, and its subgroups such as the modular group and Fuchsiangroup, based on work on automorphic...
hyperbolic Riemann surface can be defined in terms of its Fuchsian model. Suppose that the Fuchsiangroup G contains a parabolic element g. For example, the...
bounds of 5 and 10 when Γ has some nonzero odd weight modular form. A Fuchsiangroup Γ corresponds to the orbifold obtained from the quotient Γ ∖ H {\displaystyle...
algebroid Lattice (group) Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic groupFuchsiangroup Modular group Congruence...
fundamental group in the isometry group of the hyperbolic plane, which is isomorphic to PSL2(R) and this realizes the fundamental group as a Fuchsiangroup. A...
Quasi-Fuchsiangroups are obtained as quasiconformal deformations of Fuchsiangroups. By definition their limit sets are quasicircles. Let Γ be a Fuchsian group...
von Dyck group is a Fuchsiangroup, a discrete group consisting of orientation-preserving isometries of the hyperbolic plane. Triangle groups preserve...
two compact Riemann surfaces of the same genus >1 with the same quasi-Fuchsiangroup. The measurable Riemann mapping theorem shows more generally that the...
Poincaré half-plane model H of 2-dimensional hyperbolic geometry. Given a Fuchsiangroup, that is, a discrete subgroup Γ of PSL(2, R), Γ acts on H via linear...
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