Fuchsian

Fuchsian refers to objects and theories connected with the work of Lazarus Fuchs in complex analysis and differential equations, and more broadly to a class of structures arising in hyperbolic geometry. The term is used in several, related areas of mathematics, including Fuchsian groups, Fuchsian differential equations, and Fuchsian 3-manifolds, each with its own defining ideas but linked by the underlying theory of regular singularities and hyperbolic symmetry.

A Fuchsian group is a discrete subgroup of PSL(2,R) acting on the hyperbolic plane H^2 by Möbius transformations. These groups are isometries of H^2, and their quotients H^2/Γ are hyperbolic surfaces or orbifolds of finite or infinite area, depending on whether Γ is cofinite, cocompact, or has cusps. By the uniformization theorem, every hyperbolic surface can be realized as H^2/Γ for some Fuchsian group Γ. Fuchsian groups are central to the study of modular forms, Teichmüller theory, and the geometry of Riemann surfaces.

Fuchsian differential equations are linear differential equations in the complex plane whose singularities are regular. Equivalently, near each singular point, solutions have controlled growth, and the global behavior is described by a monodromy representation obtained by analytic continuation around singularities. The hypergeometric equation is a canonical example of a Fuchsian equation, having three regular singular points at 0, 1, and ∞.

In hyperbolic 3-manifold geometry, a Fuchsian manifold is a complete hyperbolic 3-manifold of the form H^3/Γ with Γ a Fuchsian group. Such manifolds are isometric to the product Σ_g × R, where Σ_g is a closed hyperbolic surface, and they appear as a symmetric special case within the broader quasi-Fuchsian landscape.