If you want to go directly to the applet, without the long-winded explanation, click here.

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 $PSL(2,\CC)$.

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.

- Generally, the left mouse button selects a parameter, the middle button moves the window, and the right mouse button zooms. If one of these functions does not make sense for the window in question, the corresponding button instead does the same function as another adjacent button. In particular:
- in the Distance to Axis window:
- left and middle chose the parameter; and
- right zooms;
- in the Trace window:
- left chooses the parameter;
- middle translates the window; and
- right zooms;
- while in the Output window:
- left and middle translate the window; and
- right zooms.
- Picking a parameter can always be done by either clicking or dragging.
- Translating is done by dragging with the appropriate button held down.
- To zoom, one drags up (to zoom in) and down (to zoom out) with the appropriate button held down. Zooming is always centered on wherever the center of the window happens to be.
- Clicking on the reset button for either panel resets its translational component and magnification.
- Clicking on the "+" button (resp., the "-" button) for any window zooms in (resp., out) by a factor of two.
- Typing a comma-separated pair of real numbers followed by RETURN into the window that displays the value of the trace moves the trace to the point with those real and imaginary parts.
- Typing a real number and RETURN in where the distance to the axis value is displayed uses that numerical value for the distance.
- Try clicking on the "decompose" button -- but then be patient, it takes a while to calculate the decomposition of the trace plane by combinatorial type of the resulting fundamental domain.

Finally, here is the applet. |

This research was supported in part by NSF grant DMS-9806408.

Jonathan Poritz (jonathan@poritz.net) |
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