Geometric Group Theory

[803396 . mosher]几何群论的一个中心主题是，有限生成的群可以通过尝试将它们分类为准等距来进行几何研究。在这个项目的一部分中，研究者将与芝加哥大学的B. Farb合作，试图证明如果G是一个有限生成的群，它是拟等长的可解群，那么G与一些可解群通过简单的代数运算密切相关，特别关注三维解流形和其他多环群的基本群。在另一部分，研究者将与M. Sageev合作，构建具有高余维但与算术格不密切相关的词双曲群的新例子，这是M. Gromov提出的一个问题。继续长期的努力，研究者将研究W. Thurston的夸张猜想的以下弱版本，即如果M是紧的3流形，则M的基本群要么是词双曲的，要么包含一个与秩2的自由阿贝尔群同构的子群。本部分将利用以前与U. Oertel合作开发的工具，以及几何群理论家最近开发的“粗代数拓扑”新工具。继续另一个长期的努力，研究者将工作表明，如果Mis是紧双曲3流形，那么只需要有限多个不同的伪anosov流来计算m的二次同调上的Thurston范数。群是几何对象对称性的数学抽象。墙纸、浴室瓷砖的对称图案，以及西班牙阿尔罕布拉宫墙壁上的瓷砖都是欧几里得对称群的例子。埃舍尔的许多画作都给出了欧几里德对称群或双曲对称群的例子。本项目将涵盖几何群论、几何及其对称群的一般研究等几个主题。通过将几何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步。研究者与Benson Farb（芝加哥大学）合作，将研究一类被称为“可解群”的特殊群，这是欧几里得对称群的一种推广。他将与同事Michah Sageev合作，研究双曲对称群的新推广。他还将继续长期致力于三维几何及其对称群的研究，以及三维动力系统的研究

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.***