Geometric Group Theory and the Topology of Aspherical Manifolds

项目负责人：Michael W. davis这是一篇关于几何群理论的研究报告，主要研究曲面的Coxeter群、Artin群和映射类群。需要解决的主要问题如下。(1)哪些考克斯特组的考克斯特图是由该组唯一确定的？哪些Coxeter群的基本生成集是由群唯一确定的（直到共轭）？哪个外自同构群是有限的？(2)所有的Artin群都是线性群吗？(3)求出与建筑物紧支上同调的公式。同样的道理也适用于一家Artin集团的Salvetti复合体。(4)确定直角Coxeter群相关的立方流形的l^2 Betti数。它们会在中维之外消失吗？这就暗示了关于奇维球体三角剖分的标志复猜想，并对图嵌入有启示。(5)发展模拟反思小组理论。这是一类类似于Coxeter群的群，由超平面排列的放大变换而来。(6)至少有三个属面的映射类群的Torelli子群是否有限生成？(7)一个词双曲群可以是由曲面束构成的曲面的基本群吗？是否所有有限表示的非双曲群都包含一个Baumslag-Solitar群或一个2阶的阿别群？群论起源于对物体对称性的研究。当这个物体具有有趣的几何结构时，人们可以使用几何技术来更好地理解对称组。这个项目涉及到在广泛的数学和物理背景下出现的群体的某些家族的研究，例如对晶体结构和DNA交织的研究。这些群与丰富而美丽的几何结构有关，这些结构使它们适合几何群论的技术。

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.