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The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element, g_2(z)=-1/z , and an element of infinite order, g_infinity(z)=z+1 . Viewing the action of the group elements on a model of the hyperbolic plane provides insight into the structure of hyperbolic 2-space. Animations provide dynamic illustrations of this action.
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