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Volume 9, Issue 3


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The Modular Group

A Finitely Generated Group with Interesting Subgroups

Paul R. McCreary
Teri Jo Murphy
Christan Carter

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, , and an element of infinite order, . 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.





*Möbius Transformations

*The Modular Group and Hyperbolic Space

*Structure of the Modular Group

*Conjugate Subgroups

*Coding Considerations

*Applications and Extensions


*Additional Material

Paul R. McCreary
Department of Mathematics
Xavier University of Louisiana

Teri Jo Murphy
Department of Mathematics
University of Oklahoma

Christan Carter
Department of Mathematics
Xavier University of Louisiana

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