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

建议：DMS-9971219PI：Ross Geoghegan摘要：Geoghegan教授从事拓扑学、几何学和群论的研究。与Robert Bieri教授(德国法兰克福)合作，他正在研究“非正曲线”空间M上群G的任意作用的新的拓扑性质。他们引入了作用在M上的n-连通性的概念，并发现这是所有此类作用空间上的一个开放条件。对于具有离散轨道的作用，他们证明了它们的条件等价于M中的点的稳定器的众所周知的有限性质，从而证明了这些性质也是开放性质。许多作者以前在一个受限案例上的工作表明，这应该是一条很有前途的研究路线，将拓扑学方法以一种新的方式引入代数。他们把目光投向了一个测试案例，SL(2，Z[1/m])在双曲平面(对于正整数m)上的Moebius作用量。他们猜想连通性的大小与m的本原因子的数量直接相关，他们还提出了证明这一点的建议。Geoghegan教授将与加拿大麦克马斯特的Andrew Nicas教授合作，继续他对参数不动点理论及其与非奇异流的关系的研究。他们希望阐明已经为这种流定义的一些K-理论挠率的几何意义。他们已经完成了该计划的一部分，但还有更多的工作要做。数学涉及到各种风格或形式的思维。其中有几何的(或视觉的)和代数的(或离散的)。例如，“离散群论”(对称性的形式研究)一直是抽象代数的一部分。但近年来，通过对这种代数进行几何化，我们对对称性的理解取得了卓有成效的进展。其中一种方法是通过“拓扑空间的基本群”的概念来实现。如果选择了明智的拓扑学空间，那么通过这个空间将“代数拓扑学”这一非常古老和深入的领域应用到正在研究的群中，就可以清楚地了解以前是不可见的和未知的群的代数。对称性的研究是数学的许多应用中的基础，特别是在物理学中。在本提案中使用这种方法是为了更好地理解群论与几何和数论的相互作用(与比耶里的部分)以及与动力学中的流动(与尼卡斯的部分)的相互作用。

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).