Topics in Discrete Groups and Geometry

离散群与几何richard Schwartz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。Schwartz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流形。当三角形群的角度发生变化时，理想的边界似乎需要进行Dehn手术。Schwartz还计划研究圆的商，基于双曲平面上在面群作用下不变的测地线模式。这个想法是为极限集建立拓扑模型，这些极限集可能与表面群变形成李群有关，然后使用该模型来研究表面群变形成李群的实际变形。广义地说，施瓦茨的研究涉及无限重复图案的几何。晶格是欧几里得空间中无限重复模式的一个例子。类似的图案存在于弯曲的空间中，通常空间的曲率允许存在更奇特和几何丰富的图案。施瓦茨感兴趣的是研究这些奇异的模式，当它们是由一个具有有限描述的简单机制产生的。例如，一个人可以在弯曲的空间里放置几面镜子，然后观察一个物体在镜子里不断反射的图像所产生的图案。核心问题是：有限的生成机制何时会导致无限的离散模式？

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?