Topics in the Geometry of Groups and Complexes

AbstractAward：DMS-0906962首席研究员：诺埃尔布拉德本研究议程的长期目标是了解群体和复合体的大尺度几何形状。具体目标包括分析高维fillinginvariants的群体，更高的填充不变量在无穷大（更高的分歧），并调查的结构ofasymptomatic锥群。其他具体的objectives包括使用组合莫尔斯理论在理解虚拟集群在某些类别的群体，并在开发一种方法的关系差距问题。主要研究者还打算比较非正曲率方法和拓扑方法（使用stablecommutator长度）证明存在的表面子群在各类双曲groups.Groups是由数学家研究对称性。一个群就是一个物体的对称性的集合。例子包括墙纸图案的几何对称性，或晶体结构，或一幅油画，或与多项式根相关的代数对称性。自19世纪以来，数学家们一直将群作为抽象代数对象进行深入研究。在20世纪80年代，M。Gromov提出把群看作几何对象，并开始推导群的几何性质和代数性质之间的深层联系。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，而是粗糙或“大尺度”。 例如，无限长的梯子和无限长的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研究大规模版本的等周问题（面积与周长问题）的群体，也是几何的粗negativelycurved群体。这些研究有助于加深我们对对称性本质的理解。

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.