If you want to go directly to the applet, without the long-winded explanation, click here.
This was a joint project with Todd Drumm; we received assistance from Robert Miner and Davide Cervone for Geomview, Adam Rosien for Java and other members of the fantastic staff of the Geometry Center at the University of Minnesota.
We were interested in using computer-aided visualization to understand the geometric and combinatorial types of fundamental domains of discrete groups of isometries of various three-dimensional geometries. For the most part, the groups we will study are cyclic. To warm up in this area, we wanted to use these tools to clarify some old results of Troels Jorgensen on the Ford fundamental domains of cyclic loxodromic subgroups $G$ of $PSL(2,\CC)$.
To describe these results, we work in the upper-halfspace model of three-dimensional hyperbolic space $H^3_\RR$, and identify its boundary with the complex numbers $\CC$. Say $G$ is generated by an element $g$ which fixes the geodesic (a vertical half-circle) from $0$ to $1$ in the complex plane. Note that $g$ is determined by its trace, which may be assumed to lie in the upper-halfplane of (another copy of) $\CC$. There exist hemispheres $I_{-n}$ and $I_n$ sitting on the complex plane with the property that $g^n$ acts as a Euclidean isometry from one to the other; these are called isometric spheres, and their intersections with the boundary $\CC$ are called isometric circles. The Ford fundamental domain for the action of $G$ on $H^3_\RR$ is the complement of the union of the interiors of the isometric spheres of all powers of the generator. There is similarly a Ford domain on the boundary of hyperbolic space consisting of the complements of the union of the interiors of the isometric circles.
Jorgensen showed in 1973 that the Ford domain $D$ of these groups on the boundary of $H^3_\RR$ is always bounded by either two, four or six circular arcs. Furthermore, he described a decomposition of the upper-halfplane of traces into regions where the boundary components of the corresponding $D$ are the circles $I_{-1}$ and $I_1$, or are bounded by arcs of the circles $I_{-m}$, $I_{-n}$, $I_m$ and $I_n$, or arcs of the circles $I_{-m}$, $I_{-n}$, $I_{-m-n}$, $I_m$, $I_n$, and $I_{m+n}$ (where, in both cases, $(m,n)=1$). In addition, Jorgensen asserted that the Ford domain in the interior of hyperbolic space could have arbitrarily many faces, although he gave no direct sequence of examples.
A brief calculation shows that to compute the Ford domain, one need only consider powers of $g$ less than an integer maxpow, depending on the trace of $g$. The existence of this bound makes it feasible to investigate the combinatorial properties of Ford domains by computational methods. We have set up tools to do this both for Ford and Dirichlet domains.
Let now $G$ be any discrete subgroup of $PSL(2,\CC)$. The Dirichlet fundamental domain for the action of $G$ on $H^3_\RR$ based at a point $x_0\in H^3_\RR$ is the set of points that are at least as close to $x_0$ as they are to any of its translates by elements of $G$. Like the Ford domain, the Dirichlet is bounded by Euclidean hemispheres centered on the boundary of the upper-halfspace model -- in fact, the limit of Dirichlet domains as the basepoint goes to infinity is exactly the Ford domain, at least for groups $G$ which do not fix infinity. The parameter space for Dirichlet domains of cyclic groups $G$ is the closed upper-halfplane (for the trace with imaginary part normalized to be non-negative) cross the non-negative real line (for the distance from the basepoint to the invariant geodesic of $G$). We have been able to show that the decomposition of this parameter space by the combinatorial type of the resulting fundamental domain is essentially just a deformation of the decomposition for Ford domains: for any fixed value of the distance from the basepoint to the invariant geodesic, we have a strikingly similar picture of the regions of the trace plane for which different isometric circles contribute to the boundary of the fundamental domain. In particular, Jorgensen's main result that every domain has either two, four or six arcs as its boundary still holds.
To study the situation on the boundary of hyperbolic space, we have written a Java applet that displays Ford and Dirichlet domains of a group after the trace of of its generator is chosen. The controls of this applet are as follows:
Warning: This applet uses Java 1.1, so if you're running an old browser, you may get errors upon loading the applet.
Finally, here is the applet. |
The above applet is actually quite useful in thinking also about Ford and Dirichlet domains in the interior of hyperbolic space -- one has only to imagine a hemisphere sitting over each disk displayed to see in one's mind's eye this three-dimensional domain. Unfortunately, Java did not have sufficient 3D display capability when this applet was first written actually to do the domains in the interior of hyperbolic space in the same way as we do on the boundary. However, using the 3D visualization software Geomview and its associated scripting module StageManager, both from the Geometry Center, we have built a tool that easily lets one examine the Ford domains inside $H^3_\RR$. This does not attach easily to Web pages, so we have assembled a small gallery of images of particularly interesting Ford domains. In particular, we illustrate by these images the surprising fact mentioned above that while the Ford domain on the boundary has two, four or six edges, in the interior of hyperbolic space the Ford domains can have arbitrarily many faces, i.e., arbitrarily many isometric spheres can contribute to the boundary of the Ford domain. In the paper we wrote on this subject, we give a complete description of the indices of the isometric spheres of which do contribute; it depends only on the $m$ and $n$ of the arcs on the boundary, via something we called the corresponding Farey sequence. See the paper on my [p]reprints page for more details.