Lattices in automorphism groups of polyhedral complexes

PI将继续研究局部紧群中的格，局部紧群是多面体复形的自同构群。李群中格的问题由来已久，但更一般拓扑群中格的基本问题仍未得到解答。提出的研究使用了各种技术来解决这些基本问题：组合、几何、代数和分析。这种广度，以及所提出问题的经典性质，意味着该提议将几个数学子领域联系在一起，特别是几何群论和格子、刚性、建筑、代数和Kac-Moody群等主题。与用于研究树格的Bass-Serre理论类似，群的复形及其覆盖理论被用作研究高维复形的自同构群中的格的有力工具。格是作为许多空间的对称的离散集合而产生的，在物理和化学中具有重要的意义。PI感兴趣的是由多面体复合体的对称性组成的格子，即由多面体粘合而成的空间。该项目解决了关于这些晶格的基本问题，并有望为经典案例提供新的见解。这项研究还将数学的几个领域联系在一起，特别是几何群论与建筑、线性代数群和卡克-穆迪理论的主题。

The PI will continue research on lattices in locally compact groups which are automorphism groups of polyhedral complexes. The subject of lattices in Lie groups has a long history, but fundamental questions about lattices in more general topological groups remain unanswered. The research proposed addresses these basic questions using a variety of techniques: combinatorial, geometric, algebraic and analytical. This breadth, and the classical nature of the questions being asked, means that the proposal connects several mathematical subfields, in particular geometric group theory and the topics of lattices, rigidity, buildings, and algebraic and Kac-Moody groups. In analogy with Bass-Serre theory, used for studying tree lattices, the theory of complexes of groups and their coverings is used as a powerful tool for studying lattices in automorphism groups of higher-dimensional complexes.Lattices arise as discrete sets of symmetries of many spaces, and are important in physics and chemistry. The PI is interested in lattices which consist of symmetries of polyhedral complexes, that is, spaces constructed by gluing together polyhedra. The project addresses basic questions about these lattices, and will hopefully give new insights into classical cases. This research also connects several areas of mathematics, in particular geometric group theory and the topics of buildings, linear algebraic groups, and Kac-Moody theory.