RUI: Geometry and Dynamics of Outer Space

提出的项目检查的几何和动力学性质的外自同构群的自由群，Out(F)，和它的同伴，Culler-Vogtmann外空间CV。第一个项目进一步推进Behrstock， Bestvina和Clay对自由群的交数的渐近性及其对某些类似于曲线复形的简单复形上Out(F)动力学的影响的研究。第二个项目进一步发展了Clay和Pettet的完全不可约自同构的新构造。这将应用于第三个项目，Clay和Pettet将开始用非对称度量法探索外层空间的稀薄部分。第四个项目旨在证明Out(F)的非虚阿贝尔子群满足“均匀-均匀”指数增长。期末计画建议让本科生检视曲面曲线复合体的子复合体的结构。克莱将与本科生一起创建一个计算机程序，用于研究自由群体，该群体汇集了几种经典和已建立的算法。图是由边连接的点的集合。具有相同数量“洞”的图可以被认为是空间中的点，称为外空间。这个空间的形状是众所周知的，但它的几何形状却不是。已知的一个令人惊讶的几何特征是空间是不对称的，这意味着两点之间的移动可能在一个方向上比另一个方向上花费更长的时间。对于任何一个爬过山的人来说，这种想法是很自然的。这个项目涉及对外层空间不对称性的分析。

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.