Topics in Discrete Groups and Geometry

离散群和几何主题

• 批准号: 0603983
• 负责人: Richard Schwartz
• 金额: $ 4.73万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603983&HistoricalAwards=false
• 关键词: Topics; Discrete; Groups; Geometry

ABSTRACTTOPICS IN DISCRETE GROUPS AND GEOMETRYRichard Schwartz plans to continue his research ingeometry and discrete groups.One of his main goals is to explore the connectionsbetween three dimensional real hyperbolic geometryand four dimensional complex hyperbolic geometry.Schwartz observed that certain complex hyperbolicdeformations of the reflection triangle groupslead to complex hyperbolic four-manifolds whoseideal boundaries are real hyperbolic three-manifolds.As the angles in the triangle group change, theideal boundary appears to undergo Dehn surgery.Schwartz also plans to study quotients of the circle,based on patterns of geodesics in the hyperbolicplane which are invariant under the action of asurface group. The idea is to make topological models for the limit sets which couldarise in connection with deformations of surfacegroups into Lie groups and then use the modelsto study actual deformations of surface groups intoLie groups.Broadly speaking, Schwartz' research deals withthe geometry of infinite repeating patterns.A crystal lattice is an example of an infiniterepeating pattern in Euclidean space. Analogouspatterns exist in curved spaces, and oftenthe curvature of the space allows for theexistence of more exotic and geometrically rich patterns.Schwartz is interested in the studying theseexotic patterns when they are generated by a simplemechanism which has a finite description.For example, one can place several mirrors ina curved space and look at the pattern generatedby the images of an object which is reflectedendlessly in the mirrors.The central question is: When does the finitemechanism of generation lead to an infinitediscrete pattern?

离散群和几何的抽象主题Richard Schwartz计划继续他的研究几何学和离散群。他的主要目标之一是探索三维真实的双曲几何和四维复双曲几何之间的联系。Schwartz观察到反射三角形群的某些复双曲变形导致复双曲四维流形，其理想边界是真实的双曲三维流形。流形。由于三角形群中的角的变化，理想的边界似乎经历Dehn手术。Schwartz还计划研究圆的几何，基于在曲面群的作用下不变的双曲平面中的测地线模式。 其思想是建立极限集的拓扑模型，这些极限集可能与表面群到李群的变形有关，然后使用这些模型来研究表面群到李群的实际变形。广义地说，Schwartz的研究涉及无限重复模式的几何。晶格是欧几里得空间中无限重复模式的一个例子。 类似的模式存在于弯曲的空间中，并且通常空间的曲率允许存在更多奇异的和几何丰富的模式。Schwartz感兴趣的是研究这些奇异的模式，当它们由具有有限曲率的简单机制产生时。例如，一个人可以在弯曲的空间里放置几面镜子，观察镜子中无反射物体的图像所产生的图案。问题是：什么时候有限的生成机制会导致无限的离散模式？