Arithmetic Groups and Tessellations of Homogeneous Spaces

这个项目的一个重点是研究齐次空间的镶嵌。也就是说，如果G/H是一个连通李群G的非紧单连通齐性空间，问题是是否存在G的一个真不连续子群D，使得轨道空间D\G/H是紧的。 L. Auslander，Y. Benoist，G. A.马古利斯河J. Zimmer和其他人。 与H合作。哦，还有A。Iozzi，PI最近在理解G是真实的秩为2的半单李群的情况方面取得了进展，包括对G = SO（2，2n）或SU（2，2n）的情况的详细研究。 PI将继续这项研究，包括真实的2级和更高的真实的级。 他还将继续他的研究行动的算术集团的圆圈，和相关的问题。这个项目研究晶体在数学空间以外的三维宇宙，我们生活在。（晶体是原子结构非常对称的材料。） 这个问题最基本的问题是决定哪些空间包含晶体，哪些不包含。 (For在这个问题上，最有趣的空间是齐次的，这意味着空间中的每个点看起来都和所有其他点完全一样。 近年来，数学家们在这个问题上取得了实质性的进展，这个项目将继续这项工作。 在晶体确实存在的情况下，该项目将研究由晶体的对称性形成的群的代数性质。

AbstractWitteOne focus of this project is the study of tessellations of homogeneous spaces. Namely, if G/H is a non-compact, simply connected homogeneous space of a connected Lie group G, the question is whether there is a properly discontinuous subgroup D of G, such that the orbit space D\G/H is compact. Some special cases were studied by L. Auslander, Y. Benoist, G. A. Margulis, R. J. Zimmer, and others. In collaboration with H. Oh and A. Iozzi, the PI has recently made progress in understanding the case where G is a semisimple Lie group of real rank two, including a detailed study of the case where G = SO(2,2n) or SU(2,2n). The PI will continue this research, both for real rank two and higher real rank. He will also continue his study of actions of arithmetic groups on the circle, and related questions.This project studies crystals in mathematical spaces other than the 3-dimensional universe that we live in. (A crystal is a material whose atomic structure is very symmetric.) The most fundamental problem in this subject is to decide which spaces contain crystals, and which do not. (For this question, the most interesting spaces are homogeneous, which means that every point of the space looks exactly like all of the other points.) Mathematicians have made substantial progress on this problem in recent years, and this project will continue the work. In cases where crystals do exist, the project will investigate the algebraic properties of the group formed by the symmetries of a crystal.