Pseudo-Aanosov automorphisms of free groups

In this thesis we present a constructive solution to Nielsen’s problem: Given a homeomorphism f of a surface S, we determine fixed points and periodic points of f. We also compute the index of fixed points. Furthermore, if f is a pseudo- Anosov homeomorphism, we illustrate the dynamics of f by finding two invariant measured foliations of S that are stretched resp. contracted by f. This solution is due to [HB92a, HB92b]. The approach is entirely algorithmic, and the appendix contains a C-program that solves the above problem for a homeomorphism that is given in terms of the induced automorphism of the fundamental group. A surface homeomorphism induces an automorphism of the fundamental group. Automorphisms induced by pseudo-Anosov homeomorphisms can be characterized in terms of algebra, so the geometric notion of pseudo-Anosov homeomorphisms has an algebraic counterpart. We analyze this interdependence. The theory is applied to homeomorphisms of surfaces with one puncture and negative Euler characteristic. The induced automorphisms of the fundamental group provide a rich source of examples, some of which are presented in detail. In addition to the algebraic information of the group automorphisms, the program also handles the topological information necessary to illustrate the dynamics of a surface homeomorphism.