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...

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...

space. Angle of parallelism Anosov flow Fuchsian group Fuchsian model Hyperbolic motion Kleinian model Models of the hyperbolic plane Pseudosphere Schwarz–Ahlfors–Pick...

analogy to those of Fuchsian models; however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs...

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 {\displaystyle...

are known as Fuchsian groups. The quotient space H2‍/‍Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic...

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...

Schwarz–Ahlfors–Pick theorem. Fuchsian group Fuchsian model Kleinian group Kleinian model Poincaré disk model Poincaré half-plane model Prime geodesic Hershel...

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...

geodesics. Consider the Poincaré half-plane model H of 2-dimensional hyperbolic geometry. Given a Fuchsian group, that is, a discrete subgroup Γ of PSL(2...

just conjugate to Fuchsian groups under conformal transformations. Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal...

orientation preserving isometries of the upper half-plane model of the hyperbolic plane. A Fuchsian group is sometimes considered as a special case of a Kleinian...

the Poincaré half-plane model of hyperbolic geometry. Riemann surfaces of negative curvature may be defined as Fuchsian models, that is, as the quotients...

Hurwitz triplet can be formed as a Fuchsian model, the quotient of the hyperbolic plane by one of these three Fuchsian groups. The Gauss–Bonnet theorem...

as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the sense of a lattice in a...

Varchenko, A. (2003). "Critical Points of Functions, sl2 Representations, and Fuchsian Differential Equations with only Univalued Solutions". Moscow Mathematical...

be obtained by Dehn surgeries on the limit manifold. Sequences of quasi-fuchsian surface groups of given genus can converge to a doubly degenerate surface...

quotient PSL(2, R) is simple. Discrete subgroups of PSL(2, R) are called Fuchsian groups. These are the hyperbolic analogue of the Euclidean wallpaper groups...

Siegel modular forms. Cusp neighborhood Extended complex upper-half plane Fuchsian group Fundamental domain Half-space Kleinian group Modular group Moduli...

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

31 October 2017. Bolibrukh AA (1995). 21st Hilbert Problem for Linear Fuchsian Systems. Amer Mathematical Society. ISBN 0-8218-0466-9. Gross DJ, Migdal...

{\displaystyle \tau } -functions for linear systems of Fuchsian type are defined below in § Fuchsian isomonodromic systems. Schlesinger equations. For the...

results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket)...

subgroup) Frieze group Wallpaper group Space group Crystallographic group Fuchsian group Modular group Congruence subgroup Kleinian group Discrete Heisenberg...

constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the...

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

Euclidean group Even and odd permutations Frieze group Frobenius group Fuchsian group Geometric group theory Group action Homogeneous space Hyperbolic...

meteorology (weather modeling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modeling (population competition)...

pairs of modular curves. Quotients of H that are compact do occur for Fuchsian groups Γ other than subgroups of the modular group; a class of them constructed...

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