If you want to go directly to the applet, without the long-winded explanation,
click here.
Visualizing Fundamental Domains
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
$PSL(2,\CC)$.
Ford Domains: Background
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.
Dirichlet Domains: Background
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.
Ford and Dirichlet Domains on the Boundary
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:
-
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.
Warning: This applet uses Java 1.1, so if you're running an old
browser, you may get errors upon loading the applet.
Ford Domains in the Interior
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.
This research was supported in part by NSF grant DMS-9806408.