Geometric Group Theory

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到拟等距来进行几何研究。在这个项目的一个部分，研究人员将与芝加哥大学的B.Farb合作，试图证明如果G是一个有限生成的群，它是一个与可解群准等距的有限生成群，那么G通过简单的代数运算与某些可解群密切相关，特别关注由三维可解流形和其他多循环群组成的基本群。在另一部分，作者将与M.Sageev合作，构造余维高但与算术格关系不密切的字双曲群的新实例，这是M.Gromov提出的一个问题。经过长期的努力，研究者将研究W.瑟斯顿的双曲猜想的下列弱版本，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M的基本群包含一个同构于2阶自由双曲群的子群。这一部分将利用以前与U Oertel合作开发的工具和群几何理论家最近开发的新的“粗代数拓扑”工具。继续另一个长期的努力，研究人员将致力于证明：如果M是紧致双曲3-流形，则只需要有限多个不同的伪Anosov流来计算M的二次同调上的瑟斯顿范数。群是几何对象的对称性的数学抽象。西班牙阿尔罕布拉宫墙上墙纸、浴室瓷砖和瓷砖的对称图案给出了欧几里德对称群的例子。埃舍尔的许多画都给出了欧几里得或双曲对称群的例子。这个项目将涵盖几何群论、几何及其对称群的一般研究的几个主题。通过将年龄计量学的大尺度特性与其对称群的大尺度特性相匹配，人们可以利用几何来研究群，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。这位研究人员将与本森·法布(芝加哥大学)合作，研究一类特殊的群，即欧几里得对称群的推广，即所谓的“可解群”。与他的同事Michah Sageev合作，他将研究双曲对称群的新推广。他还将继续长期努力，了解三维几何及其对称群，以及研究三维动力系统。

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.***