Topics in the Geometry of Groups and Complexes

群与复形几何专题

• 批准号: 0906962
• 负责人: Noel Brady
• 金额: $ 15.45万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906962&HistoricalAwards=false
• 关键词: Topics; Geometry; Groups; Complexes

AbstractAward: DMS-0906962Principal Investigator: Noel BradyThe long-term objective of this research agenda is to understandthe large scale geometry of groups and complexes. Specificobjectives include the analysis of higher dimensional fillinginvariants of groups, higher filling invariants at infinity(higher divergence), and investigating the structure ofasymptotic cones of finitely presented groups. Other specificobjectives include the use of combinatorial Morse theory inunderstanding virtual fibering in certain classes of groups, andin developing an approach to the relator gap problem. Theprincipal investigator also intends to compare non-positivecurvature methods and topological methods (using stablecommutator length) for proving the existence of surface subgroupsin various classes of hyperbolic groups.Groups are used by mathematicians to study symmetry. A group isjust a collection of symmetries of an object. Examples includethe geometric symmetries of a wallpaper pattern, or of a crystalstructure, or of an Escher painting, or the algebraic symmetriesassociated to roots of polynomials. Mathematicians have studiedgroups intensively as abstract algebraic objects since the 19thcentury. In the 1980's M. Gromov proposed that we consider groupsas geometric objects, and began to derive deep connectionsbetween the geometric and the algebraic properties of groups. Onetheme which emerged from Gromov's work is that the geometricproperties which have deep algebraic consequences are not localproperties, but rather coarse or "large scale". For example, andinfinite ladder and an infinite straight line are not locallyalike, but are large scale alike, and their symmetry groups willhave many algebraic similarities. The PI investigates large scaleversions of isoperimetric problems (area versus perimeter lengthproblems) in groups, and also the geometry of coarsely negativelycurved groups. These investigations help deepen our understandingof the nature of symmetry.

摘要奖：DMS-0906962首席研究员：诺埃尔·布雷迪这项研究议程的长期目标是理解基团和络合物的大规模几何结构。具体目标包括分析群的高维填充不变量，无限(高散度)的高维填充不变量，以及研究有限表示群的渐近圆锥的结构。其他具体目标包括使用组合莫尔斯理论来理解某些类群中的虚拟纤维，以及开发一种解决相对人缺口问题的方法。主要研究人员还打算比较非正曲率方法和拓扑法(使用稳定的交换子长度)来证明各种类型的双曲群中曲面子群的存在。群被数学家用来研究对称性。群就是一个对象的对称性的集合。例子包括墙纸图案的几何对称性，或晶体结构的几何对称性，或埃舍尔油画的几何对称性，或与多项式的根有关的代数对称。自19世纪以来，数学家一直将群作为抽象的代数对象进行深入研究。20世纪80年代，S·M·格罗莫夫提出将群视为几何对象，并开始推导出群的几何性质与代数性质之间的深刻联系。从格罗莫夫的工作中出现的一个主题是，具有深刻代数后果的几何性质不是局部性质，而是粗略或“大尺度”。例如，无限阶梯和无限直线不是局部相似的，而是大尺度相似的，它们的对称群将有许多代数相似之处。PI在组中研究大尺度的等周问题(面积与周长的问题)，以及粗负曲线群的几何。这些研究有助于加深我们对对称性本质的理解。