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...

In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If...

Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic...

is called a quasi-Fuchsian group. When the Jordan curve is a circle or a straight line these are just conjugate to Fuchsian groups under conformal transformations...

hyperbolic plane. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A Fuchsian group that preserves orientation...

hyperbolic plane). Generalising the example of the modular group a Fuchsian group 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 Fuchsian group. Every hyperbolic...

developments of automorphic forms other than modular forms. The case of Γ a Fuchsian group had already received attention before 1900 (see below). The Hilbert...

fundamental group in the isometry group of the hyperbolic plane, which is isomorphic to PSL2(R) and this realizes the fundamental group as a Fuchsian group. A...

Henrik Abel in 1827. Bianchi group Classical modular curve Fuchsian group J-invariant Kleinian group Mapping class group Minkowski's question-mark function...

negative curvature may be defined as Fuchsian models, that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the...

neighborhood Extended complex upper-half plane Fuchsian group Fundamental domain Half-space Kleinian group Modular group Moduli stack of elliptic curves Riemann...

algebroid Lattice (group) Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic group Fuchsian group Modular group Congruence...

NEC group which contains only orientation-preserving elements is called a Fuchsian group, and any non-Fuchsian NEC group has an index 2 Fuchsian subgroup...

that can be subsets of Jordan curves Lakes of Wada Quasi-Fuchsian group, a mathematical group that preserves a Jordan curve Jordan (1887). Kline, J. R...

surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important...

Poincaré and Klein introduced the group of Möbius transformations, and its subgroups such as the modular group and Fuchsian group, based on work on automorphic...

hyperbolic Riemann surface can be defined in terms of its Fuchsian model. Suppose that the Fuchsian group G contains a parabolic element g. For example, the...

isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface). The topological type of X...

Poincaré half-plane model H of 2-dimensional hyperbolic geometry. Given a Fuchsian group, that is, a discrete subgroup Γ of PSL(2, R), Γ acts on H via linear...

bounds of 5 and 10 when Γ has some nonzero odd weight modular form. A Fuchsian group Γ corresponds to the orbifold obtained from the quotient Γ ∖ H {\displaystyle...

listed as a grave of honour of the State of Berlin. He is the eponym of Fuchsian groups and functions, and the Picard–Fuchs equation. A singular point a of...

point group, Schoenflies notation Discrete group Euclidean group Even and odd permutations Frieze group Frobenius group Fuchsian group Geometric group theory...

field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects...

Quasi-Fuchsian groups are obtained as quasiconformal deformations of Fuchsian groups. By definition their limit sets are quasicircles. Let Γ be a Fuchsian group...

limit manifold. Sequences of quasi-fuchsian surface groups of given genus can converge to a doubly degenerate surface group, as in the double limit theorem...

universal property. Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on...

index 2. Torsion-free normal subgroups of the (2,3,7) triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath...

two compact Riemann surfaces of the same genus >1 with the same quasi-Fuchsian group. The measurable Riemann mapping theorem shows more generally that the...

the Riemann surface is conformally equivalent to H/Γ where Γ is a Fuchsian group of Möbius transformations. A fundamental domain for Γ is given by a...