RUI: Geometry and Dynamics of Outer Space

Allegheny College, Meadville PA

Investigators

Abstract

The proposed projects examine the geometry and dynamical properties of the outer automorphism group of a free group, Out(F), and its ompanion, the Culler-Vogtmann Outer space CV. The first project furthers Behrstock, Bestvina and Clay's investigation of the asymptotics of intersection numbers for free groups and its implications to the dynamics of Out(F) on certain simplicial complexes analogous to the curve complex. The second project further develops Clay and Pettet's new construction of fully irreducible automorphisms. This has applications for the third project where Clay and Pettet will begin to explore the thin part of the Outer space with the asymmetric metric. The fourth project aims to show that nonvirtually abelian subgroups of Out(F) satisfy "uniform-uniform'' exponential growth. The final project proposes to involve undergraduates in examining the structure of subcomplexes of the curve complex for a surface. Together with undergraduates, Clay will create a computer program for studying free groups that assembles several classical and established algorithms. A graph is a collection of points connected by edges. Graphs with the same number of "holes" can be considered themselves as points in a space, called Outer space. The shape of this space is well-known, but its geometry is not. One surprising geometric feature that is known is that the space is asymmetric, meaning traveling between two points may take longer in one direction than the other. This notion is very natural to anyone who has hiked up a mountain. This project involves an analysis of the asymmetries of Outer space.