Free Groups, Their Automorphism Groups, Applications

The free group F n with n generators and its automorphism group Φ n , studied in combinatorial group theory, provide an algebraic non-commutative scheme for quasiperiodic systems. Homomorphisms from F 2 to the groups SU(2) and SU(1,1) are considered. A particular geometry, inspired by Fricke and Klein, is developed for these groups. The elements of Φ2 in this geometry are generated by elementary transformations of triangles on orbit spaces. The Fibonacci system is examined as an example. Matrix invariants and their geometric interpretation are studied beyond the usual trace map. The groups yield symmetry principles for spin and electron dynamics in quasiperiodic structures.