Applications of Algebraic Topology to Geometric Group Theory, Parametrized Fixed Point Theory and Dynamics

代数拓扑在几何群论、参数化不动点理论和动力学中的应用

• 批准号: 9971219
• 负责人: Ross Geoghegan
• 金额: $ 4.65万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971219&HistoricalAwards=false
• 关键词: Applications; Algebraic; Topology; Geometric; Group

Proposal: DMS-9971219PI: Ross GeogheganAbstract: Professor Geoghegan is working at the interface of topology, geometryand group theory. In collaboration with Professor Robert Bieri(Frankfurt, Germany) he is investigating new topological properties ofan arbitrary action by a group G on a "non-positively curved" space M.They have introduced the idea of n-connectedness of the action over Mand have found that this is an open condition on the space of all suchactions. For actions with discrete orbits they have identified theirconditions as equivalent to well-known finiteness properties ofstabilizers of points in M thus proving that these are also openproperties. Previous work of many authors on a restricted casesuggests that this should be a promising line of investigation, bringingtopological methods into algebra in a new way. They have set theirsights on a test case, the Moebius action of SL(2, Z[1/m]) on thehyperbolic plane (for a positive integer m). They conjecture that theamount of connectvity is directly related to the number of primefactors of m, and they propose to prove this among other things. Incollaboration with Professor Andrew Nicas (McMaster, Canada) ProfessorGeoghegan will continue his study of parametrized fixed point theoryand its relationship to non-singular flows. They hope to elucidate thegeometrical meaning of some K-theoretic torsions which have beendefined for such flows. They have already achieved part of thatprogram but more remains to be done.Mathematics involves various styles or forms of thought. Among theseare the geometrical (or visual) and the algebraic (or discrete). Forexample, "discrete group theory" (the formal study of symmetries) hastraditionally been part of abstract algebra. But fruitful advances inour understanding of symmetries have been made in recent years throughgeometrizing this kind of algebra. One way of doing this is throughthe notion of "fundamental group of a topological space". If thetopological space is chosen wisely, much about the algebra of the groupwhich was previously invisible and unknown becomes clear throughapplying the very old and deep field of "algebraic topology" to thegroup being studied, via this space. The study of symmetries is basicin many applications of mathematics, particularly to physics. Thisapproach is being used in the present proposal to better understand theinterplay of group theory with geometry and number theory (in the partwith Bieri) and with flows in dynamics (in the part with Nicas).

摘要：Geoghegan教授主要从事拓扑、几何和群论的交叉研究。在与Robert Bieri教授（法兰克福，德国）的合作中，他正在研究群G在“非正弯曲”空间m上的任意作用的新拓扑性质。他们引入了在Mand上的作用的n连通性的思想，并发现这是所有此类作用空间上的开放条件。对于具有离散轨道的作用，他们已经确定了它们的条件等同于众所周知的M中点的稳定子的有限性质，从而证明了这些也是开放性质。许多作者以前在一个有限情况下的工作表明，这应该是一个有前途的研究路线，将拓扑方法以一种新的方式引入代数。他们将目光放在了一个测试用例上，即双曲平面（对于正整数m）上SL（2, Z[1/m]）的莫比乌斯作用。他们推测，连通性的数量与m的素因子的数量直接相关，他们打算在其他事情中证明这一点。纪勤教授将与加拿大麦克马斯特的Andrew Nicas教授合作，继续研究参数化不动点理论及其与非奇异流的关系。他们希望阐明为这种流动所定义的一些k理论扭转的几何意义。他们已经实现了该计划的一部分，但仍有更多的工作要做。数学涉及各种思维方式或形式。其中包括几何（或视觉）和代数（或离散）。例如，“离散群论”（对对称性的正式研究）传统上是抽象代数的一部分。但是，近年来，通过对这类代数的几何化，我们对对称性的理解取得了丰硕的进展。一种方法是通过“拓扑空间的基本群”的概念。如果拓扑空间的选择是明智的，那么通过将“代数拓扑”这个古老而深刻的领域应用到被研究的群体中，关于群体的代数的许多以前不可见和未知的东西就会变得清晰起来。对称的研究是许多数学应用的基础，特别是在物理学中。这种方法被用于本提案中，以更好地理解群论与几何和数论（与Bieri的部分）以及与动力学中的流动（与Nicas的部分）的相互作用。