Mathematical Sciences: Topics in Low-Dimensional Topology and Geometric Group Theory

9504946本研究项目主要集中在格罗莫夫意义上的负曲率。主要研究人员(与U·Oertel合作)证明了“分层定理”，该定理指出，对于紧致胞腔-复形X，两种选择中的一种恰好成立：要么X是Gromov意义下的负曲线，要么存在到X的具有某些性质的二维分层映射，即分层具有欧拉特征为零的横向度量，并且从叶子的泛覆盖到X的映射是最小面积的。这个项目的主要目的是利用分层定理来研究3-流形M的“弱双曲猜想”，即M是Gromov意义下的负曲线当且仅当M的基本群没有Z+Z子群。这一猜想的强形式一直是三维流形理论中最大的挑战之一。根据Cannon最近关于刻划双曲3-流形群的工作和Gabai关于双曲流形的拓扑刚性问题的工作，弱猜想应该是证明强猜想的重要一步；如果这些不同的项目能够完成，那么双曲猜想就会被证明。这个研究项目还涉及其他几个主题，特别是：理解自动机和双自动机群的性质；映射类群的几何群论；三维流形上伪Anosov流的构造和性质。群论是E·伽罗瓦在1700年由S发明的关于对称性的数学研究，尽管它起源于几何，但在它的历史上大部分时间都是抽象代数的主题。从M·格罗莫夫、J·坎农、W·瑟斯顿等人在20世纪70年代的S和80年代的S的工作开始，几何群论的新兴领域使群论回归了它的起源。格罗莫夫在1983年向国际数学运动学大会发表讲话时提出的基本问题是取一个“群”(一个抽象描述的对称集合)，并理解它的几何形状。作为格罗莫夫程序的一个特例，假设S是一个三维空间，很像我们居住的空间，假设G是S的某个对称群；在这种情况下，瑟斯顿在20世纪70年代末的《S》中猜想，G的几何应该归入一系列特定的类。在大多数情况下，瑟斯顿的猜想表明，G应该有“双曲”几何，这是一种描述相对论中三维空间如何融入四维时空的几何。目前研究计划的主旨是试图证明瑟斯顿的“双曲猜想”。D.Gabai，J.Cannon和U.Oertel的研究人员最近的工作提出了一个证明双曲猜想的三步法，这个项目涉及其中的一步，即所谓的“弱双曲猜想”。虽然这是一个雄心勃勃的项目，但研究人员预计它将是富有成效的；如果它完全成功，如果其他两个步骤像坎农和加拜所说的那样完成，那么双曲猜想将被证明。***

9504946 Mosher This research project is focussed on negative curvature in the sense of Gromov. The principal investigator (working jointly with U. Oertel) has proved a "Lamination Theorem" which says that for a compact cell-complex X, exactly one of two alternatives holds: either X is negatively curved in the sense of Gromov, or there is a 2-dimensional lamination mapping to X with certain properties, namely the lamination has a transverse measure of Euler characteristic zero, and the map from the universal cover of a leaf to X is least area. The primary aim of this project is to use the Lamination Theorem to investigate the "weak hyperbolization conjecture" for a 3-manifold M, which says that M is negatively curved in the sense of Gromov if and only if the fundamental group of M has no Z+Z subgroup. The strong form of this conjecture has been one of the greatest challenges in 3-manifold theory. The weak conjecture should be a significant step in proving the strong conjecture, in light of recent work of Cannon on characterizing hyperbolic 3-manifold groups, and work of Gabai on the topological rigidity problem for hyperbolic manifolds; if these various projects can be completed, then the hyperbolization conjecture would be proved. This research project is also concerned with several other topics, in particular: understanding properties of automatic and biautomatic groups; the geometric group theory of mapping class groups; constructions of and properties of pseudo-Anosov flows on 3-manifolds. Group theory, the mathematical study of symmetry invented by E. Galois in the 1700's, has for much of its history been a subject of abstract algebra, despite its geometric origins. Starting with work of M. Gromov, J. Cannon, W. Thurston and others in the 1970's and 1980's, the newly emergent field of geometric group theory has returned group theory to its origins. The basic problem, proposed by Gromov in his 1983 address to the International Congress of Math ematics, is to take a "group" (an abstractly described collection of symmetries) and to understand its geometry. As a particular case of Gromov's program, suppose that S is a 3-dimensional space, much like the space that we inhabit, and suppose that G is a certain group of symmetries of S; in this situation Thurston conjectured in the late 1970's that the geometry of G should fall into a list of specific classes. In most cases Thurston's conjecture says that G should have "hyperbolic" geometry, a type of geometry that describes how 3-dimensional space fits into 4-dimensional space-time in relativity theory. The main thrust of the current research project is an attempt to prove Thurston's "hyperbolization conjecture." Recent work of D. Gabai, J. Cannon, and the investigator wih U. Oertel suggests a three-step approach to proving the hyperbolization conjecture, and this project is concerned with one step, the so-called "weak hyperbolization conjecture." While this is an ambitious project, the investigator expects it to be productive; if it is completely successful, and if the other two steps are finished as the work of Cannon and of Gabai suggests, then the hyperbolization conjecture will be proved. ***