Editing Modular group (section)

==Maps of the torus==
The group {{math|GL(2, '''Z''')}} is the linear maps preserving the standard lattice {{math|'''Z'''<sup>2</sup>}}, and {{math|SL(2, '''Z''')}} is the orientation-preserving maps preserving this lattice; they thus descend to [[self-homeomorphism]]s 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 [[Homotopy#Isotopy|isotopic]] to a map of this form. The algebraic properties of a matrix as an element of {{math|GL(2, '''Z''')}} correspond to the dynamics of the induced map of the torus.