PSL(2,R)

PSL(2,R)
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	PSL(2,R)
Type	Projective special linear group
Field	Real numbers
Related	SL(2,R), SO(2,1), Möbius group

PSL(2,R) is the group of 2×2 real matrices with unit determinant modulo its center, acting faithfully by orientation-preserving isometries of the hyperbolic plane and by Möbius transformations on the extended real line. It is a non-compact, simple Lie group of real rank one that arises in connections with Bernhard Riemann's uniformization ideas, Henri Poincaré's tessellations, and the modern theories of Andrew Wiles-related modular forms and Grigori Perelman-style geometric structures. PSL(2,R) plays a central role across Felix Klein's Erlangen program, the theory of Atiyah–Singer index theorem applications, and the study of discrete subgroups such as Hecke operators-related arithmetic groups.

Definition and Basic Properties

PSL(2,R) is defined from Évariste Galois's linear groups as SL(2,R)/{±I}, where SL(2,R) is the group studied by Camille Jordan and Issai Schur. As a real Lie group it is connected, has dimension three, and its Lie algebra is isomorphic to sl(2,R) studied by Sophus Lie and Wilhelm Killing. The group is locally isomorphic to SO(2,1), appears in the classification of simple Lie groups by Élie Cartan, and exhibits properties crucial to the development of the Langlands program and the Selberg trace formula pioneered by Atle Selberg.

Algebraic Structure and Isomorphisms

Algebraically, PSL(2,R) contains one-parameter subgroups corresponding to exponentials in sl(2,R) analyzed by David Hilbert and Élie Cartan. Up to isomorphism it is the identity component of the group of orientation-preserving isometries of the hyperbolic plane and is closely related to matrix groups appearing in Hermann Weyl's representation theory. Isomorphisms link PSL(2,R) to the group of Möbius transformations considered by August Ferdinand Möbius and studied in the context of Riemann surfaces by Bernhard Riemann and Riemann–Roch theorem applications. Its centerless nature was used in classification problems by Emmy Noether and Richard Brauer.

Action on the Hyperbolic Plane and Möbius Transformations

PSL(2,R) acts by fractional linear transformations on the upper half-plane model of the hyperbolic plane, a perspective developed by Henri Poincaré and exploited in Felix Klein's study of automorphic functions. This action preserves the hyperbolic metric central to Grigory Margulis's rigidity theorems and to the Mostow rigidity theorem of G. D. Mostow. The same transformations appear in the theory of Modular group congruence subgroups studied by Hecke and Goro Shimura, and underlie classical results used by Srinivasa Ramanujan and Carl Ludwig Siegel in modular form investigations.

Subgroups and Classification (Elliptic, Parabolic, Hyperbolic)

Elements of PSL(2,R) are classified as elliptic, parabolic, or hyperbolic following a taxonomy employed by Poincaré and refined in the study of Fuchsian groups by Lars Ahlfors and Lipman Bers. Elliptic elements fix a point in the hyperbolic plane as in Johannes Kepler-inspired geometric symmetry problems; parabolic elements fix a single boundary point appearing in Gauss-related continued fraction dynamics; hyperbolic elements have two fixed boundary points and generate axial translations used by Helmut Hasse and André Weil in arithmetic geometry. Discrete subgroups such as cofinite Fuchsian groups, including arithmetic examples related to Richard Dedekind and Carl Friedrich Gauss, give rise to tessellations studied by John Conway and H. S. M. Coxeter.

Representation Theory and Covering Groups

The unitary dual of PSL(2,R) was classified in foundational work by Harish-Chandra and Israel Gelfand, with principal series and discrete series representations discovered by Harish-Chandra and Atle Selberg. The double cover SL(2,R) and higher metaplectic covers enter via André Weil's metaplectic representation and in the construction of theta functions used by Sergiu Rădulescu and Harvey Cohn. Representation-theoretic techniques link to the Langlands correspondence studied by Robert Langlands and to spectral theory applications in the work of Peter Sarnak and Dennis Sullivan.

Applications in Geometry, Dynamics, and Number Theory

PSL(2,R) underpins hyperbolic geometry developments central to William Thurston's program and the proof strategies of Grigori Perelman for the Poincaré conjecture. In ergodic theory it appears in celebrated results by Marian Rees-style mixing and by Furstenberg in rigidity contexts; in homogeneous dynamics it is pivotal in Margulis's work on unipotent flows and the Oppenheim conjecture. Number-theoretic applications include the theory of modular forms exploited by Andrew Wiles in proof elements of the Taniyama–Shimura conjecture and by Yuri Manin in algebraic cycles; trace formula methods of Atle Selberg and James Arthur link spectral data to arithmetic counts. In mathematical physics PSL(2,R) symmetry appears in conformal field theory studied by Alexander Zamolodchikov and in integrable systems investigated by Ludwig Faddeev.

Category:Lie groups Category:Hyperbolic geometry Category:Representation theory