The Giraud procedure for G(4,4,4;5)

# The Giraud procedure for G(4,4,4;5)

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 W0 of generators, and repeat the following procedure. For each set Wi, find the Dirichlet domain FWi bounded by the bisectors B(p0,w p0), w∈Wi. Each 2-face of FWi 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 Wi+1 by adding to Wi all group elements of the form gh-1, g,h∈Wi, 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 FW0.

Then you go from Wi to Wi+1 by clicking on the 'Giraud step' button. The computations to get W1 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