Geometric Group Theory via Geometric Combinatorics

这个项目的目的是通过构造适当的复形，将几何群论学者通常研究的各种群(如Artin群、单相关群、小消去群、字双曲群、充分扭面配对3-流形的基本群等)置于标准一般理论(如分段非正曲率欧几里德空间或Garside结构)的范围内。例如，对于度量小消去群，对于充分扭面配对的三维流形的基本群，对于单相关子群，对于构造二维共形CAT(0)复形，对于任意Artin群，都出现了构造高维非正曲立方体复形的策略。几何和计数组合学在结构本身以及它们的主要性质的建立中起着突出的作用。在每一种情况下，所构建的复合体和方法本身都是新的和创新的，其关键特性正在建立过程中。早期的迹象表明，这些复合体具有几何和组合结构，这些结构将解决诸如一相关群的相干性和Artin群的字问题的解决等长期存在的猜想。该项目位于几何/组合群论和几何/计数组合学之间的交界处。前者研究与几何对象(如它们的对称组)有关的代数结构，而后者可以粗略地定义为只用有限数量的数据来描述的事物的研究。这正是计算机可以做的数学类型。计算数学似乎远远脱离了几何方面的考虑，但有越来越多的组合现象，最好地可以被视为关于光滑空间曲率的事实的有限类比。这个项目的主要目标是使用这些通过计算发现的组合现象来构造具有几何/拓扑结构的络合物，然后解释观察到的原始基团的代数行为。

The goal of this project is to bring various classes of groupscommonly studied by geometric group theorists (such as Artin groups,one-relator groups, small cancellation groups, word-hyperbolic groups,fundmental groups of ample twisted face pairing 3-manifolds, etc.)within range of one of the standard general theories (such aspiecewise Euclidean spaces of nonpositive curvature, conformalnonpositive curvature, or Garside structures) by constructingappropriate complexes on which they act. For example, there areemerging strategies for constructing high-dimensional nonpositivelycurved cube complexes for metric small cancellation groups as well asfor fundamental groups of ample twisted face pairing 3-manifolds, forconstructing 2-dimensional conformally CAT(0) complexes forone-relator groups, and for constructing Garside-like structures forarbitrary Artin groups. Geometric and enumerative combinatorics playa prominent role in the constructions themselves as well as in theestablishment of their major properties. In each case, the complexesconstructed and the approaches themselves are new and innovative andtheir key properties are in the process of being established. Earlyindications are that these complexes carry geometric and combinatorialstructures which would resolve such longstanding conjectures as thecoherence of one-relator groups, and the solution of the word problemfor Artin groups.This project lies at the interface between geometric/combinatorialgroup theory and geometric/enumerative combinatorics. The formerstudies algebraic structures associated with geometric objects (suchas their group of symmetries) while the latter can be roughly definedas the study of things which can be described using only a finiteamount of data. This is precisely the type of mathematics thatcomputers can do. Computational mathematics might seem far removedfrom geometric considerations, but there is a growing collection ofcombinatorial phenomena which can best be viewed as finite analoguesof facts about the curvature of smooth spaces. The primary goal ofthis project is to use these computationally discovered combinatorialphenomena to construct complexes which carry a geometric/topologicalstructure which then explain the observed algebraic behavior of theoriginal groups.