Geometric Group Theory

几何群论

• 批准号: 9803396
• 负责人: Lee Mosher
• 金额: $ 9.34万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-09-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803396&HistoricalAwards=false
• 关键词: Geometric; Group; Theory

9803396Mosher A central theme of geometric group theory is that finitely generatedgroups can be studied geometrically by attempting to classify them up toquasi-isometry. In one part of this project, the investigator, workingjointly with B. Farb of the University of Chicago, will attempt to showthat if G is a finitely generated group that is quasi-isometric to asolvable group, then G is very closely related to some solvable group bysimple algebraic operations, with special attention to fundamental groupsof 3-dimensional solv-manifolds and other polycyclic groups. In anotherpart, joint with M. Sageev, the investigator will work on constructingnew examples of word hyperbolic groups that have high codimension but arenot closely related to arithmetic lattices, a problem posed by M. Gromov.Continuing a long-term effort, the investigator will study the followingweak version of W. Thurston's Hyperbolization Conjecture, namely that ifM is a compact 3-manifold, then the fundamental group of M is either wordhyperbolic or else contains a subgroup isomorphic to a free abeliangroup of rank 2. This part will make use of tools developed in previousjoint work with U. Oertel as well as new tools of ``coarse algebraictopology'' developed recently by geometric group theorists. Continuinganother long-term effort, the investigator will work on showing that if Mis a compact hyperbolic 3-manifold, then only finitely many differentpseudo-Anosov flows are needed to compute Thurston's norm on the secondhomology of M. Groups are the mathematical abstraction of symmetries of geometricobjects. Symmetry patterns of wallpaper, bathroom tiles, and thetilings on the walls of the Alhambra in Spain give examples of Euclideansymmetry groups. Many of Escher's paintings give examples either ofEuclidean or of hyperbolic symmetry groups. This project will coverseveral topics in geometric group theory, the general study of geometriesand their symmetry groups. By matching the large scale properties of ageometry with the large scale properties of its symmetry groups, one canuse geometries to study groups and vice versa, a very fruitful pairingthat has led to many recent mathematical advances. The investigator,working with Benson Farb (U Chicago), will study a special class ofgroups known as ``solvable groups,'' a generalization of Euclideansymmetry groups. Working with his colleague Michah Sageev, he willinvestigate new generalizations of hyperbolic symmetry groups. He willalso continue work on long term efforts to understand 3-dimensionalgeometries and their symmetry groups, as well as to study 3-dimensionaldynamical systems.***

小行星9803396 几何群论的一个中心主题是，可以通过尝试将它们分类为拟等距来几何地研究生成群。 在这个项目的一部分，研究者，与B合作。Farb等人试图证明，如果G是一个可解群的拟等距生成群，那么G通过简单的代数运算与某个可解群密切相关，特别注意三维解流形的基本群和其他多环群。 另一部分与M. Sageev，研究者将致力于构造具有高余维但与算术格没有密切关系的词双曲群的新例子，这是M.继续长期的努力，研究者将研究W. Thurston的双曲化猜想，即如果M是紧致的3-流形，则M的基本群要么是字双曲的，要么包含一个同构于秩为2的自由阿贝尔群的子群。 这一部分将利用以前与美国联合开发的工具。Oertel以及几何群理论家最近发展的“粗糙代数拓扑学”的新工具。 继续另一个长期的努力，研究者将致力于证明，如果M是一个紧凑的双曲3-流形，那么只需要2000个不同的伪Anosov流来计算M的第二同调上的Thurston范数。 群是几何对象对称性的数学抽象。 墙纸、浴室瓷砖和西班牙阿尔汉布拉墙上的瓷砖的对称图案给出了欧几里得对称群的例子。 许多布泽尔的绘画给出的例子无论是欧几里德或双曲对称群。 本计画将涵盖几何群论、几何的一般研究及其对称群的几个主题。 通过将几何学的大尺度性质与其对称群的大尺度性质相匹配，人们可以用几何学来研究群，反之亦然，这是一种非常富有成效的配对，导致了许多最近的数学进展。 调查员，与本森Farb（U芝加哥），将研究一类特殊的群称为“可解群”，一个广义的欧几里德对称群。 与他的同事Michah Sageev合作，他将研究双曲对称群的新推广。 他还将继续长期致力于理解三维几何及其对称群，以及研究三维动力系统。