Punctured Torus Explorer Explore the whirly, curly world of 2-Generator Kleinian groups
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Punctured Torus Explorer Description

This application allows you to independently set the two complex values which determine a punctured torus, and then draws the limit set of the covering group, conceived of as acting on hyperbolic 3-space + plane at infinity. There are options for controlling how many group elements are calculated; and for controlling the rendering, such as whether it should be drawn as the Riemann sphere or flat; in the former case, one can display part of the pleated surface within hyperbolic space corresponding to the group. You can also display fixed points and words associated to the almost parabolic elements in the group, which play an important role in characterizing the limit set. Finally, once you set one of the parameters, it attempts to detemine and display the boundary of the Teichmuller space for the other parameter, but this feature does not work reliably. Punctured Torus Explorer is a small but effective utility that was designed in order to help you create and view 2-Generator Kleinian groups. You can easily customize the parameters and animate the output by dragging it around. Since Punctured Torus Explorer is completely written in the Java programming language there should be no problems in running it on several systems.

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