Explore: Fuchsian Groups And Their Generalizations

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

groups Thompson's group F CAT(0) groups Arithmetic groups Automatic groups Fuchsian groups, Kleinian groups, and other groups acting properly discontinuously

A classical theorem of Henri Poincaré constructs Fuchsian groups as hyperbolic reflection groups generated by reflections in the edges of a geodesic

made by Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, who studied cases involving irregular singularities. A Fuchsian system is the system of linear differential

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

have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket) and the modular group in particular. The connection

Hilbert's problems and their solvers. Natick, Mass: A.K. Peters. ISBN 978-1-56881-141-3. Thiele, Rüdiger (2005). "On Hilbert and his twenty-four problems"

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

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

quasiconformal groups". Journal d'Analyse Mathématique. 46: 318–346. doi:10.1007/BF02796595. Gabai, Davis (1992). "Convergence groups are Fuchsian groups". Annals