The applet below lets you explore the Dirichlet domain for the image of a nice representation of the (4,4,4)-triangle group into PU(2,1), which turns out to be a cocompact lattice (the representation is essentially determined by the requirement that the word 1232 be mapped to an element of order five).

Each step of the procedure refines the set of group elements needed to bound a fundamental domain, according to the following rule.

Start from a set W_{0} of generators, and repeat the
following procedure. For each set W_{i}, find the Dirichlet
domain F_{Wi} bounded by the bisectors
B(p_{0},w p_{0}), w∈W_{i}. Each 2-face of
F_{Wi} is associated to a pair of group elements g
and h, so that the 2-face is on the intersection of the two
corresponding bisectors ĝ and ĥ.
We obtain W_{i+1} by adding to W_{i} all group
elements of the form gh^{-1}, g,h∈W_{i}, whenever
the intersection of ĝ and ĥ is not totally geodesic (and
then discard the elements whose corresponding face has empty
interior).

A more detailed description of the procedure can be found here.

The applet performs this procedure, and lets you visualize
the various 2-faces of the Dirichlet domain at each stage. If the
applet runs, in the beginning you should see a gray egg on the
screen. This is a picture of one 2-face of the Dirichlet domain
F_{W0}.

Then you go from W_{i} to W_{i+1} by clicking on
the 'Giraud step' button. The computations to get W_{1} are
very quick, but at each stage from that point on, the Giraud step take
from a dozen of minutes to a number of hours (there are many places
where the applet could be improved...)

If you are ready to let the applet run for about 15 minutes, you will see that after step 2, the Dirichlet domain is compact. If you are motivated and patient enough, you may notice that after the 4-th step, the procedure stabilizes, and you will have a list of the 2-faces of a fundamental domain for the group.

Note that this is a test version, and it probably has a number bugs (feel free to mail me a copy of your Java console if you have trouble with it). Some useful info about the progress of the computations appears in the java console.

Created by Martin Deraux