Research in geometric group theory

近年来，Coxeter群及其相关群（如Artin群）和相关空间（如建筑物）的研究在几何群论中变得越来越重要。PI（Davis）和Co-PI（Januszkiewicz）计划计算这些群和空间的各种类型的上同调。他们还将研究确定复流形的分支覆盖是否是非正弯曲的问题。 这与对Artin群的K（pi，1）猜想的可能攻击有关。 PI计划与Boris Okun一起计算群环系数的上同调以及Artin群、Bestvina-Brady群和无限群的图积的L2上同调。 合作PI，连同他的合作者在波兰，计划工作的问题，推广他的概念单纯非正曲率。非正曲率涉及领域以外的纯数学，从机器人（通过配置空间）统计力学。 考克斯特群是纯数学的许多领域的中心，从几何和拓扑到动力系统，再到数论和李群理论。Artin群的研究是Artin辫子群的推广。 辫子群也已经在不同的数学领域，例如纽结理论和代数几何。 该研究将对非正曲率领域、Coxeter群和Artin群等相关领域产生影响。

In recent years the study of Coxeter groups and their relatives (e.g. Artin groups) and related spaces (e.g. buildings) have become increasingly important in geometric group theory. The PI (Davis) and Co-PI (Januszkiewicz) plan to compute various types of cohomology for these groups and spaces. They also will study the problem of determining whether branched covers of complex manifolds are nonpositively curved. This is related to a possible attack on the K(pi, 1) Conjecture for Artin groups. Together with Boris Okun the PI plans to compute the cohomology with group ring coefficients as well as the L2 cohomology of Artin groups, Bestvina-Brady groups and graph products of infinite groups. The Co-PI, together with his collaborators in Poland, plans to work on the problem of generalizing his notion of simplicial nonpositive curvature.Nonpositive curvature relates to areas outside pure mathematics ranging from robotics (via configuration spaces) to statistical mechanics. Coxeter groups are central in many areas of pure mathematics ranging from geometry and topology to dynamical systems to number theory and the theory of Lie groups. The study of Artin groups are a generalization of Artin's braid groups. The braid groups also have been in different areas of mathematics, for example, knot theory and algebraic geometry. The proposed research will have an impact in the area of nonpositive curvature, in the areas of Coxeter groups and Artin groups and other related areas.