RUI: Geometry and Dynamics of Outer Space

RUI：外层空间的几何和动力学

• 批准号: 1006898
• 负责人: Martin Serra
• 金额: $ 9.43万
• 依托单位: Allegheny College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006898&HistoricalAwards=false
• 关键词: RUI; Geometry; Dynamics; Outer; Space

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.

拟议的项目检查的几何和动力学性质的外自同构群的自由群，出（F），和它的companion，Culler-Vogtmann外层空间CV。 第一个项目进一步Behrstock，Bestelf和克莱的调查的渐近的交叉数的自由团体和其影响的动态Out（F）的某些单纯复合类似的曲线复杂。 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。 这在第三个项目中有应用，克莱和佩泰特将开始用非对称度规探索外层空间的薄部分。 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。 最后一个专题建议让大学生参与研究曲面的曲线复形的子复形的结构。 克莱将与本科生一起创建一个用于研究自由群的计算机程序，该程序集合了几个经典的和已建立的算法。 具有相同数量的“洞”的图可以被认为是空间中的点，称为外层空间。 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，这意味着在两点之间旅行可能在一个方向上比另一个方向花费更长的时间。 这种想法对任何一个爬山的人来说都是很自然的。 该项目涉及对外层空间不对称性的分析。