Geometric Group Theory, Surface Dynamics

and the Pseudo-Anosov Homeomorphism

Professor Handel

(preliminary draft from an NSF grant abstract)

The pseudo-Anosov homeomorphism is one of the most important concepts in
the areas of mathematics known as three-dimensional geometry, three-dimensional topology, and
two-dimensional dynamical systems. The study of pseudo-Anosov homeomorphisms began in the
mid-1970's. During the decade that followed, a great deal of progress was made in analyzing their
behavior. Since then progress has slowed, in part because only the more difficult problems remain.
Chief among these problems is an understanding of the "forcing partial order," which, roughly
speaking, describes the way in which simple two dimensional systems change into chaotic ones.
I use extensive computer calculations and techniques from other
areas of mathematics, especially that of geometric group theory to investigate this problem.

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