Topics in Discrete Groups and Geometry

离散群和几何理查德·施瓦茨计划继续他在几何和离散群方面的研究。他的主要目标之一是探索三维实双曲几何和四维复双曲几何之间的联系。施瓦茨观察到反射三角形群的某些复双曲变形导致复双曲四流形，其理想边界是实双曲三维流形。随着三角形群中角度的变化，理想边界似乎进行了Dehn手术。Schwartz还计划基于双曲面群作用下不变的双曲平面上的大地平面线图案来研究圆的商。其思想是为与表面群到Lie群的变形有关的极限集建立拓扑模型，然后使用该模型来研究表面群到Lie群的实际变形。从广义上说，Schwartz的研究涉及无限重复图案的几何。晶格是欧氏空间中无限迭代图案的一个例子。类似的图案存在于弯曲的空间中，空间的曲率往往允许存在更奇异的和几何丰富的图案。Schwartz对研究这些奇异图案感兴趣，因为它们是由具有有限描述的简单机构产生的。例如，一个人可以在弯曲的空间中放置几面镜子，并观察由镜子中没有树枝反射的对象的图像生成的图案。中心问题是：生成的有限项机制何时会导致无限离散的图案？

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?