Geometric Group Theory and 3-Manifolds

9803316Stallings正在研究与格鲁什科定理有关的问题，其中涉及到一些关于曲面的微妙问题。最近的一个版本是关于紧连通曲面T到一个楔形空间A和B的映射，该映射在基本群上是满射的；如果边界曲线全部映射到一侧或另一侧，则映射同伦于沿连通部分将曲面分割成两个凹陷的映射，可能除了长度为2的映射外。他还对有关双曲群、图的割点和3-流形的Heegaard分裂的问题感兴趣。这些自然会影响他的研究生，他们的想法和计划也会影响他。一位名叫Kim Whittlesey的学生解决了一个有趣的问题，即只由平凡元素和伪Anosov映射组成的映射类群的子群的存在性。另一名学生正在研究与自由群有关的算法问题，正在深入研究自由群中的方程理论。另一名学生对使用计算机发现几何对象感兴趣，另一名学生已经开始更详细地研究自动机和双曲群理论。助学金的大部分是给学生提供部分资助的。群论起源于对几何或代数对象的对称性的研究。几何可以被软化成拓扑学，这产生了一些有趣的新想法；M·格罗莫夫是这一概念的发明者之一(1980年的S)，自从他的第一部作品以来，它已经得到了相当大的扩展。事后看来，高斯(约1800年)、庞加莱(约1900年)、怀特黑德(1930年‘S)的著作中都出现了关于群体的几何思想。三维流形的性质特别受基本群的影响，相反，许多单纯关于群的结果也受到了这些拓扑思想的启发。这种方法已经理解甚至发现了数学其他领域的几个逻辑和组合方面。在这个领域有许多优秀的研究人员，也有许多现在可以接近的问题。*

9803316Stallings Stallings is working on matters connected with Grushko's Theorem, with certain delicate questions about surfaces involved. One recent version ofthis concerns a map of a compact connected surface T into a wedge of spacesA and B, which is surjective on fundamental groups; if the boundary curvesall map to one side or the other, except possibly for one which maps withlength two, the map is homotopic to one that splits the surface into twoparts along a connected piece. He is also interested in matters concerninghyperbolic groups and cut vertices of graphs and Heegaard splittings of3-manifolds. These naturally influence his graduate students; and theirideas and plans influence him too. One student, Kim Whittlesey, has solvedan interesting problem concerning the existence of a subgroup of the mapping class group that consists only of the trivial element together with pseudo-Anosov maps. Another student is working on algorithmic mattersconcerning free groups and is getting deeply into the theory of equationsin free groups. Another student is interested in using the computer todiscover things about geometric objects, and another has started gettinginto the theory of automatic and hyperbolic groups in some detail. Themajor portion of the grant is for the partial support of students. Group theory started out as the study of symmetries of geometric oralgebraic objects. The geometry can be softened into topology, and thisproduces some interesting new ideas; M. Gromov was one of the originators of this (1980's), and it has expanded considerably since his first work. In hindsight, one can see that geometric ideas about groups have occurred in work by Gauss (circa 1800), Poincare (circa 1900), Whitehead (1930's). The nature of 3-dimensional manifolds is especially influenced by thefundamental group, and conversely many results simply about groups have been stimulated by these topological ideas. Several logical andcombinatorial aspects of other areas of mathematics have been understoodor even discovered by this approach. There are many good researchers inthis area and many problems that are now approachable.***