 Such an element acts as a squeeze mapping of the Euclidean plane, and the corresponding element of PSL(2,R) acts as a translation of the hyperbolic plane and as a Lorentz boost on Minkowski space.Hyperbolic elements of the modular group act as Anosov diffeomorphisms of the torus.Hyperbolic elements are conjugate into the 2 component group of standard squeezes × ±"I"