11 - Crossing boundaries 365 to 372

Summary

Adding the letter ‘c’ to our language

One can grow a little tired of babbling abba—bAABB—aBaa—Baba…. Certainly, language should get far more interesting if we allow c, and then d, and perhaps even e! What we mean is that we should extend our considerations to groups with more than two generators. The algorithm for enumerating the tree of words hardly changes: with a, b and c, we have a six-letter alphabet (because we have to include the inverses A, B and C). In a free group, we must of course avoid cC and Cc; in non-free groups, one would use an automaton to enumerate only the acceptable words in the tree.

Formulas for the matrices for interesting groups with more than two generators are slightly harder to come by. It is true that we can perhaps conjugate the group to obtain formulas with some special symmetry, but in general we have to expect that groups with more than two generators will exhibit much greater variety than two-generator ones.

One way to find formulas is to return to our original framework of Schottky groups and try to find interesting ways to pair up circles in the plane. This is precisely what we did to make the glowing limit set picture on p. xviii; it comes from a group with three generators, arranged as in Figure 11.5.