kleinian, a tool for visualizing Kleinian groups

It’s been a while since I last blogged; the reason, of course, is that I felt that I couldn’t post anything new before completing my series of posts on Kähler groups; but I wasn’t quite ready to write my last post, because I wanted to get to the bottom of a few analytic details in the notorious Gromov-Schoen paper. I am not quite at the bottom yet, but maybe closer than I was; but I’m still pretty far from having collected my thoughts to the point where I can do them justice in a post. So I’ve finally decided to put Kähler groups on the back burner for now, and resume my usual very sporadic blogging habits.

So the purpose of this blog post is to advertise that I wrote a little piece of software called kleinian which uses the GLUT tools to visualize Kleinian groups (or, more accurately, interesting hyperbolic polyhedra invariant under such groups). The software can be downloaded from my github repository at

https://github.com/dannycalegari/kleinian

and then compiled from the command line with “make”. It should work out of the box on OS X; Alden Walker tells me he has successfully gotten it to compile on (Ubuntu) Linux, which required tinkering with the makefile a bit, and installing freeglut3-dev. There is a manual on the github page with a detailed description of file formats and so on.

One nice feature of the program is that the user just has to give semigroup generators for their (semi)-group, and a finite list of (hyperbolic) triangle orbits; the program then computes the Cayley graph out to some (user-specified) depth, applies the resulting set of transformations to the triangles, and renders the result. The code is available, and is licensed under the GPL, and I actively encourage anyone who wants to fork it and develop it into a more powerful tool to do so.

A few examples of output are:

universal cover of a genus 3 handlebody

universal cover of the fiber of the fibration of the figure 8 knot complement

space with Sierpinski carpet limit set invariant by super-ideal simplex reflection group

I wrote this program mainly just to produce some nice figures for a recent talk I gave at U Chicago to first-year graduate students; the talk itself can be downloaded from my webpage here. If you download this program, and enjoy using it, I would be very grateful to get feedback, or just to hear about your experience.