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.

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 the following 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, here for more details.

Trace = $1+i$, top view

Trace = $1+i$, bottom view

Trace = $.5333+.95i$, top view

Trace = $.5333+.9i$, bottomw view

Trace = $1.3+.3i$, top view

Trace = $1.3+.3i$, bottom view

Trace = $1.5+.5i$, top view

Trace = $1.5+.5i$, bottom view

Trace = $.4+.4i$, top view

Trace = $.4+.4i$, bottom view

Trace = $.2+.2i$, top view

Trace = $.2+.2i$, bottom view

Trace = $.05+.05i$, top view, from far away

Trace = $.05+.05i$, bottom view, from far away

Trace = $.05+.05i$, side view, from far away

Trace = $.05+.05i$, top view, from up close

Trace = $.05+.05i$, bottom view, from up close

Trace = $.05+.05i$, angled bottom view, up close, random colors

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