Geometric Group Theory and the Topology of Aspherical Manifolds

几何群论与非球面流形拓扑

• 批准号: 0104026
• 负责人: Michael Davis
• 金额: $ 22.11万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104026&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Topology; Aspherical

AbstractAward: DMS-0104026Principal Investigator: Michael W. DavisThis is a proposal for research in geometric group theoryfocusing on Coxeter groups, Artin groups, and mapping classgroups of surfaces. The main problems to be addressed are thefollowing.(1) For which Coxeter groups is the Coxeter diagram uniquelydetermined by the group? For which Coxeter groups is thefundamental generating set uniquely determined (up toconjugation) by the group? For which Coxeter groups is the outerautomorphism group finite?(2) Are all Artin groups linear groups? (3) Find a formula for the cohomology with compact supports of abuilding. Likewise for the Salvetti complex of an Artin group.(4) Determine the l^2 Betti numbers of cubical manifoldsassociated to right-angled Coxeter groups. Do they vanish outsidethe middle dimension? This would imply the Flag ComplexConjecture concerning triangulations of odd-dimensional spheresand has implications for graph embeddings.(5) Develop a theory of mock reflection groups. This is a class ofgroups similar to Coxeter groups which arise as transformationsof blow-ups of hyperplane arrangements.(6) Is the Torelli subgroup of the mapping class groups of asurface of genus at least three finitely generated?(7) Can a word hyperbolic group be the fundamental group of asurface by surface bundle? Must all finitely presented non wordhyperbolic groups contain a Baumslag-Solitar group or an abeliangroup of rank two?Group theory arises from the study of symmetries of anobject. When this object has an interesting geometric structure,one can use geometric techniques to better understand the groupof symmetries. This project involves the study of certainfamilies of groups which arise in a broad range of mathematicaland physical contexts, such as the study of crystal structuresand the intertwining of DNA. These groups are associated to richand beautiful geometric structures which lend themselves to thetechniques of geometric group theory.

摘要奖：DMS-0104026主要研究者：Michael W.戴维斯这是一个建议的研究几何群论集中在考克斯特集团，阿廷集团，和映射类集团的表面。需要解决的主要问题如下。(1)哪些Coxeter群的Coxeter图是由群决定的？哪一个Coxeter群的基本生成集是由群唯一确定的（直到共轭）？哪些Coxeter群的外自同构群是有限的？(2)所有的Artin群都是线性群吗？ (3)求出了建筑物具有紧支集的上同调公式。对于Artin群的Salvetti复形也是如此。(4)求直角Coxeter群的三次流形的l ^2 Betti数。它们会消失在中维度之外吗？这将意味着关于奇维球面三角剖分的旗复猜想，并对图嵌入有影响。(5)建立一个模拟反射组的理论。这是一类群类似于Coxeter群出现作为transformationsblow-ups的超平面安排。(6)亏格曲面的映射类群的Torelli子群是否至少生成三个？(7)一个字双曲群能成为曲面丛的基本群吗？所有的非双曲群都必须包含Baumslag-Solitar群或秩为2的Abeliang群吗？群论起源于对物体对称性的研究。当这个物体有一个有趣的几何结构时，人们可以使用几何技术来更好地理解对称群。该项目涉及研究在广泛的南极和物理背景下出现的某些群体家族，例如研究晶体结构和DNA的缠绕。这些群与丰富美丽的几何结构相关联，这些几何结构有助于几何群论的技巧。