Maps of The Torus
The group GL(2,Z) is the linear maps preserving the standard lattice Z², and SL(2,Z) is the orientation-preserving maps preserving this lattice; they thus descend to self-homeomorphisms of the torus (SL mapping to orientation-preserving maps), and in fact map isomorphically to the (extended) mapping class group of the torus, meaning that every self-homeomorphism of the torus is isotopic to a map of this form. The algebraic properties of a matrix as an element of GL(2,Z) correspond to the dynamics of the induced map of the torus.
Read more about this topic: Modular Group
Famous quotes containing the words maps of and/or maps:
“The faces of most American women over thirty are relief maps of petulant and bewildered unhappiness.”
—F. Scott Fitzgerald (1896–1940)
“And now good morrow to our waking souls,
Which watch not one another out of fear;
For love all love of other sights controls,
And makes one little room an everywhere.
Let sea-discoverers to new worlds have gone,
Let maps to other, worlds on worlds have shown,
Let us possess one world; each hath one, and is one.”
—John Donne (1572–1631)