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If you're interested in Hyperbolic Geometry or Computational Group Theory I have a thesis on Automatic Coset Systems I'd love to share with you. It's got Hyperbolic Geometry, Finite State Automata, Coset Enumeration, Automatic Groups and other nice stuff in it. It includes the proof that a quasiconvex subgroup of a hyperbolic group is strongly geodesically coset automatic and therefore Short-Lex coset automatic, together with an algorithm (implemented in about 15,000 lines of C++, available on request) that will determine the word acceptor and multipliers (and hence a coset enumeration) for any coset automatic coset space.
In fact it cheats - the algorithm will determine any finite state automaton from a sufficiently large sample of its language - and I use a simple Knuth-Bendix reduction to feed it with irreducible words (i.e. things not containing left hand sides of rewrite rules).
In general, if it's possible to enumerate the cosets of a finitely generated subgroup of a finitely presented group, this algorithm will find you a way to do it. But you will need to bring a sleeping bag.
Using the enumeration automata above, John Parker, Greg Mcshane, André Rocha and I were able to enumerate cosets in many hyperbolic groups, producing some fascinating orbit plots. Some of this work appeared in the Journal of Experimental Mathematics Volume 3.
If you're not interested, you can just look at the pictures, many of which were constructed at The Geometry Center at the University of Minnesota based on work done at the Mathematics Institute of the University of Warwick, where my PhD supervisor was Derek Holt.