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

这个研究项目的重点是格罗莫夫意义上的负曲率。首席研究员（与U. Oertel合作）证明了一个“层积定理”，该定理表明，对于紧致细胞复合体X，只有两种选择之一成立：X在Gromov意义上是负弯曲的，或者存在到X的二维层积映射具有某些性质，即层积具有欧拉特征零的横向度量，并且从叶子的普遍覆盖到X的映射面积最小。本课题的主要目的是利用层积定理研究3流形M的“弱夸张猜想”，该猜想表明M在Gromov意义上是负弯曲的，当且仅当M的基本群没有Z+Z子群。这个猜想的强形式一直是三流形理论中最大的挑战之一。鉴于Cannon最近关于双曲3流形群的刻画工作和Gabai关于双曲流形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；如果这些不同的工程都能完成，那么夸张猜想就被证明了。本研究项目还涉及其他几个主题，特别是：理解自动和双自动群体的性质；映射类群的几何群论3流形上伪anosov流的构造与性质。群论是17世纪由伽罗瓦（E. Galois）发明的关于对称性的数学研究，尽管它起源于几何，但在其历史的大部分时间里，它一直是抽象代数的主题。从20世纪七八十年代M. Gromov、J. Cannon、W. Thurston等人的工作开始，新兴的几何群论领域使群论回到了它的起源。1983年，格罗莫夫在国际数学大会上提出的基本问题是，取一个“群”（一个抽象描述的对称集合），并理解它的几何形状。作为格罗莫夫规划的一个特例，假设S是一个三维空间，就像我们所居住的空间一样，假设G是S的一组对称性；在这种情况下，瑟斯顿在20世纪70年代末推测G的几何应该属于一系列特定的类。在大多数情况下，瑟斯顿猜想认为G应该具有“双曲”几何，这是一种描述相对论中三维空间如何适应四维时空的几何。当前研究项目的主要目的是试图证明瑟斯顿的“夸张猜想”。D. Gabai、J. Cannon和U. Oertel的研究人员最近的工作提出了一个三步法来证明夸张猜想，而这个项目只涉及一步，即所谓的“弱夸张猜想”。虽然这是一个雄心勃勃的项目，研究者希望它是富有成效的；如果它是完全成功的，如果其他两个步骤都完成了Cannon和Gabai的工作，那么夸张猜想将被证明。＊＊＊

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. ***