Computational Kleinian Groups

In 1979, Mumford recruited me and a year later McMullen to computationally explore the space of free two-generator Kleinian groups. We were the first to produce computer graphics programs to draw limit sets of arbitrary free two-generator groups, and to map the space of these groups, as parametrized by the traces of their generators. With the aid of these programs, we made many remarkable pictures and discovered several patterns in the limit sets. We also made the first picture of the `cuspy curve' which is the fractal boundary of one of the simplest nontrivial Teichmüller spaces, namely, Maskit's construction of the once-punctured torus space, where each point $\mu$ corresponds to a Kleinian group generated by the two Möbius transformations

$$a_\mu(z) = \mu+\frac1z b(z) =z+2$$

Mumford had visions of a book for the general public that would entice them with pictures of extraordinary symmetry and lead them into the beautiful mathematics of discrete groups and the geometry of Riemann surfaces. This was a far more ambitious goal than Curt or I were able or willing to undertake at the time, and while I generally agreed we should do this, for some years I had no resources to help carry this out.

In 1987 at Göttingen, I spent some time figuring out exactly what was the mathematics that we were supposed to make available to the general public. In that endeavor, I wrote most of a manuscript [Wri87b] explaining the definition of Maskit's Teichmüller space of once-punctured tori, which was essentially an extraction of a concrete example from Maskit's very general papers [Mas68,Mas74], explained the calculation of the cusps on this boundary, showed the pattern of the `circle chain' in a cusp group, and gave a precise asymptotic formula for the shape of the cusps in the boundary. The asymptotic formula was based on hypotheses that took many years to establish, and the precise formula is still not entirely proved. The continuity of the boundary was established in [Min99] and the cardioid shape of the cusps is proved in [Miy03].

I gave this manuscript to Caroline Series when she visited Göttingen, and because of her expertise in the combinatorics of Kleinian groups and Riemann surfaces and ultimately three-manifolds she was the ideal person to prove and explain much of the patterns of circle chains that appear in the limit sets of cusp groups. A handsome stream of research has emerged from those early works of Series and Keen.

When I returned to Stillwater, events prevented me from really finishing the $T_{1,1}$ manuscript, even though ultimately it is perhaps my most cited work. I did complete a new revision of the limit set and boundary drawing programs called kleinian. In 1994, while at the Institute for Advanced Study, I met with Mumford and he had convinced Caroline to help us write what would eventually become Indra's Pearls [MSW02]. I wrote an architecture of the book at IAS in 1995 and work began in earnest around 1996. Thus, began a vagabond life for me to mostly Providence and Warwick to work on what was a far more massive undertaking than I had even guessed as a student. Still the book contains accounts of limit set calculations and boundary tracing algorithms that appear nowhere else. It has been gratifying to see talented amateurs around the world take up these programs and produce their own pictures. A shining example is Jos Leys' work at www.josleys.com, including thousands of limit set pictures.

Over the course of time, I have used kleinian to contribute dozens of pictures to mathematical works of others. See for example the boundary of $T_{1,1}$ occurring in Milnor's account of McMullen's work [LLMM99].

After Indra's Pearls, I have started to produce more detailed accounts of the mathematics behind the pictures. A complete account of the web of circles that appears in the limit set of any maximal cusp group is presented in [Wri05]. This also describes the similarity between the geometry of maximal cusp groups and that of the coherent spiral hexagonal circle packings (also known as Doyle packings) treated in [BDS94]. We give a conjectural asymptotic formula for the similarity constants of Doyle packings that is an analogue of the cardioid shape of cusps.

The other short papers [MW04,Wri04] both are outcomes of study of the patterns of maximal cusp groups. The graphic Double Cusp Group presented in [Wri04] is a result of my realization that there is a simple direct enumeration procedure for the disks in certain maximal cusp groups that allows a coherent coloring. The coloring vividly illustrates the extra symmetries that occur in a `tangential' geometric limit of Kleinian groups. This graphic is currently on display in Paris in the exhibition Mathématíques et Arts (see http://www.hermay.org/IHP/). I am preparing a manuscript that explains this alternative coset enumeration for these cusp groups.

Although I have often been torn by the demands of two vastly different scholarly endeavors, I have a few more works to finish on my computations in kleinian groups, including a `revisited' edition of my old manuscript [Wri87b] and a supplement of more details on programming kleinian groups, which exist now as an untidy collection of emails to Indra's Pearls readers around the world. Then I hope to participate in the next generation of work on algebraic groups and algebraic number fields heralded by Bhargava.