Relationship To Hyperbolic Geometry
See also: PSL2(R)The modular group is important because it forms a subgroup of the group of isometries of the hyperbolic plane. If we consider the upper half-plane model H of hyperbolic plane geometry, then the group of all orientation-preserving isometries of H consists of all Möbius transformations of the form
where a, b, c, and d are real numbers and ad − bc = 1. Put differently, the group PSL(2,R) acts on the upper half-plane H according to the following formula:
This (left-)action is faithful. Since PSL(2,Z) is a subgroup of PSL(2,R), the modular group is a subgroup of the group of orientation-preserving isometries of H.
Read more about this topic: Modular Group
Famous quotes containing the words relationship to, relationship and/or geometry:
“Poetry is above all a concentration of the power of language, which is the power of our ultimate relationship to everything in the universe.”
—Adrienne Rich (b. 1929)
“Whatever may be our just grievances in the southern states, it is fitting that we acknowledge that, considering their poverty and past relationship to the Negro race, they have done remarkably well for the cause of education among us. That the whole South should commit itself to the principle that the colored people have a right to be educated is an immense acquisition to the cause of popular education.”
—Fannie Barrier Williams (1855–1944)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”
—J.G. (James Graham)