Algorithms for subgroups and automorphisms of right-angled Artin Groups

一个有限表现群是直角阿廷群（RAAG），如果生成元之间的唯一关系是某些生成元对可交换。 私家侦探将研究RAAG的子群结构和自同构群。 这个项目的主要目标是开发用于分析RAAGs的算法，这些算法将推广经典算法，包括Nielsen约简，Stallings折叠，列约简和Whitehead算法。 目标应用程序包括一个算法，用于测试成员在自同构组轨道RAAGs，和限制版本的子组成员问题。 第二个目标是定义RAAG自同构群作用的组合和拓扑对象，并使用这些动作来更好地理解自同构群。 第三个目标是证明关于RAAG的自同构群的重要子群的同调有限性结果;项目的这一部分将特别关注自由群的自同构群。直角阿廷群（RAAG）是一种代数结构，其中所有方程都是断言某些元素对可交换的方程（形式为“x*y=y*x”的方程）的结果。 私家侦探将研究一般RAAG的结构，特别是这些对象的子结构，以及它们的对称性（自同构）。 RAAG及其子结构和对称性是几何群论的重要研究对象;这些群包括自群论起源以来一直在研究的特殊情况。 自由阿贝尔群（例如坐标n空间中的整数点的格）是RAAG的例子;它们的自同构群是矩阵群，许多关于它们的结构和对称性的问题可以用熟悉的矩阵技术来回答，例如行和列约简。 一类称为自由群的群也是RAAG的例子。 Nielsen约化和Whitehead算法是20世纪20年代和30年代的组合算法，用于分析自由群的某些方面，Stallings折叠是20世纪70年代的相关图论技术。 该项目的主要目标是构建这些经典的自由群算法和矩阵技术，将同时适用于所有RAAG的共同推广，或证明不存在这样的推广。

A finitely-presented group is a right-angled Artin group (RAAG) if the only relations between the generators are that some pairs of generators commute. The P.I. will study the subgroup structure and automorphism groups of RAAGs. The main goal of this project is to develop algorithms for analyzing RAAGs; these algorithms will generalize classical algorithms including Nielsen reduction, Stallings folding, column reduction, and Whitehead's algorithm. Goal applications include an algorithm for testing membership in automorphism group orbits in RAAGs, and restricted versions of the subgroup membership problem. A second goal is to define combinatorial and topological objects on which automorphism groups of RAAGs act, and to use these actions to better understand the automorphism groups. A third goal is to prove homological finiteness results about important subgroups of automorphism groups of RAAGs; this part of the project will focus on automorphism groups of free groups specifically.A right-angled Artin group (RAAG) is a type of algebraic structure in which all equations are consequences of equations asserting that certain pairs of elements commute (equations of the form "x*y=y*x"). The P.I. will research the structure of general RAAGs, specifically the substructures of these objects, and their symmetries (automorphisms). RAAGs and their substructures and symmetries are important objects of study in geometric group theory; these groups include special cases that have been studied since the origins of group theory. Free abelian groups (for example the lattice of integer points in coordinate n-space) are examples of RAAGs; their automorphism groups are matrix groups, and many questions about their structure and symmetry can be answered using familiar matrix techniques such as row and column reduction. A class of groups called free groups are also examples of RAAGs. Nielsen reduction and Whitehead's algorithm are combinatorial algorithms from the 1920's and 1930's for analyzing certain aspects of free groups, and Stallings folding is a related graph-theoretical technique from the 1970's. The main goal of the project is to construct common generalizations of these classical free group algorithms and matrix techniques that would simultaneously apply to all RAAGs, or to prove that no such generalizations exist.