Lattices in automorphism groups of polyhedral complexes

PI将继续研究多面体配合物的自同构群——局部紧群中的格。李群中的格的研究已经有很长的历史了，但是关于更一般拓扑群中的格的基本问题仍然没有答案。研究提出了解决这些基本问题，使用各种技术：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，意味着该提案连接了几个数学子领域，特别是几何群论和晶格、刚性、建筑、代数和Kac-Moody群的主题。与研究树格的Bass-Serre理论类似，群及其复盖的复形理论是研究高维复形自同构群中的格的有力工具。晶格是许多空间的离散对称集合，在物理和化学中很重要。PI感兴趣的是由多面体配合物的对称性组成的晶格，即由多面体粘合在一起构成的空间。该项目解决了关于这些格子的基本问题，并有望为经典案例提供新的见解。这项研究还连接了数学的几个领域，特别是几何群论和建筑、线性代数群和Kac-Moody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.