Algorithms for subgroups and automorphisms of right-angled Artin Groups

有限表示群是直角Artin群(RAAG)，如果生成元之间的唯一关系是某些生成元对可交换。P.I.将研究RAG的子群结构和自同构群。这个项目的主要目标是开发分析RAG的算法；这些算法将推广经典算法，包括Nielsen归约、Stallings折叠、列归约和Whitehead算法。目标应用包括测试RAG中自同构群轨道的成员资格的算法，以及子群成员资格问题的受限版本。第二个目标是定义RAG的自同构群作用于其上的组合和拓扑对象，并使用这些作用来更好地理解自同构群。第三个目标是证明RAAG的自同构群的重要子群的同调有限性结果；该项目的这一部分将具体地关注自由群的自同构群。直角Artin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的结果(形式为x*y=y*x的方程)。P.I.将研究一般RAG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)。RAG及其子结构和对称性是几何群论中的重要研究对象，这些群包括自群论诞生以来一直被研究的特殊情况。自由阿贝尔群(例如，坐标n空间中的整点的格子)是RAAG的例子；它们的自同构群是矩阵群，并且许多关于它们的结构和对称性的问题可以使用熟悉的矩阵技术来回答，例如行和列归约。一类称为自由组的组也是RAG的例子。尼尔森归约和怀特黑德算法是20世纪20年代S和30年代S提出的用于分析自由群某些方面的组合算法，而Stallings折叠是70年代S提出的一种相关图论技术。该项目的主要目标是构造这些经典自由群算法和矩阵技术的共同推广，它们将同时适用于所有RAG，或者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.