Geometric Group Theory via Geometric Combinatorics

这个项目的目标是将几何群论中常用的各种群（如Artin群、单关系子群、小消去群、字双曲群、充分扭曲面配对3-流形的基本群等）引入到几何群论中。在一个标准的一般理论（如非正曲率，conformalnonpositive曲率，或Garside结构的分段欧几里得空间）的范围内，通过构建适当的复合物，他们的行为。 例如，对于度量小消去群和充分扭曲面配对3-流形的基本群，构造高维非正曲立方复形，对于单关系子群构造2维共形CAT（0）复形，以及对于任意Artin群构造Garside类结构，都出现了新的策略。 几何和枚举组合学在构造本身以及建立它们的主要性质中起着突出的作用。 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，这些复杂的几何和combinatorialstructures，这将解决这些长期的aceturtures作为theherence的一个relator组，和解决的word problemfor阿廷groups.This项目之间的接口位于几何/combinatorialgroup理论和几何/枚举组合。 前者研究与几何对象相关的代数结构（如它们的对称群），而后者可以粗略地定义为研究仅用有限数量的数据就可以描述的事物. 这正是计算机可以做的数学类型。 计算数学似乎远离几何考虑，但有越来越多的组合现象，可以最好地被看作是有限的类比的事实曲率的光滑空间。 该项目的主要目标是利用这些计算发现的组合现象来构建具有几何/拓扑结构的复合体，然后解释原始群的观察到的代数行为。

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.