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

提案：DMS-9971219 PI：Ross Geoghegan摘要：Geoghegan教授正在研究拓扑学、几何学和群论的接口。 在合作教授罗伯特Bieri（法兰克福，德国），他正在调查新的拓扑性质ofan任意行动的一组G的“非积极弯曲”空间M.他们介绍了想法n-连通的行动超过M并发现，这是一个开放的条件空间的所有suchactions。 对于具有离散轨道的作用，他们已经确定了他们的条件等价于著名的M中点的稳定器的有限性性质，从而证明了这些也是开性质. 以前的工作，许多作者对一个限制casesugests，这应该是一个有前途的调查路线，bringtopological方法代数在一个新的方式。 他们把目光放在一个测试案例上，即SL（2，Z[1/m]）在双曲平面上的Moebius作用（对于正整数m）。 他们猜想连通性的数量与m的素因子的数量直接相关，他们打算证明这一点。 Geoghegan教授将与Andrew Nicas教授（加拿大麦克马斯特）合作，继续研究参数化不动点理论及其与非奇异流的关系。 他们希望能阐明为这类流动定义的某些K-理论挠率的几何意义。 他们已经完成了这个计划的一部分，但还有更多的工作要做。数学涉及各种风格或形式的思想。 其中包括几何（或视觉）和代数（或离散）。 例如，“离散群理论”（对称性的形式研究）已经成为抽象代数的一部分。 但近年来，通过将这种代数几何化，我们对对称性的理解已经取得了丰硕的进展。 这样做的一种方法是通过“基本组的拓扑空间”的概念。 如果拓扑空间是明智的选择，许多关于代数的群，这是以前看不见的和未知的变得清晰，通过应用非常古老和深刻的领域“代数拓扑”的群正在研究，通过这个空间。 对对称性的研究是许多数学应用的基础，特别是在物理学中。这种方法被用在本提案中，以更好地理解群论与几何和数论（在与Bieri的部分中）以及与动力学中的流动（在与Nicas的部分中）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).