Geometric Group Theory and 3-Manifolds

9803316Stallings Stallings正在研究与格鲁什科定理有关的问题，涉及到一些关于曲面的微妙问题。最近的一个版本涉及到一个紧连通曲面T到空间a和B的楔形的映射，它在基本群上是满射的；如果边界曲线都映射到一边或另一边，除了可能映射长度为2的地图，则该地图与沿着连接部分将表面分成两部分的地图是同伦的。他对双曲群和图的切顶点以及3流形的切分也很感兴趣。这些自然会影响他的研究生；他们的想法和计划也影响着他。一个学生，Kim Whittlesey，解决了一个有趣的问题，关于映射类群的子群的存在性，这个子群只由平凡元素和伪anosov映射组成。另一名学生正在研究与自由群有关的算法问题，并正在深入研究自由群的方程理论。另一名学生对使用计算机发现几何物体感兴趣，另一名学生已经开始深入研究自动和双曲群的理论。补助金的主要部分是用于学生的部分支持。群论最初是研究几何或代数对象的对称性。几何图形可以被软化成拓扑结构，这产生了一些有趣的新想法；M. Gromov是这个（1980年代）的创始人之一，自他的第一部作品以来，它已经大大扩展了。事后看来，我们可以看到高斯（大约1800年）、庞加莱（大约1900年）和怀特黑德（20世纪30年代）的研究中出现了关于群体的几何思想。三维流形的性质特别受到基本群的影响，相反，这些拓扑思想激发了许多关于群的简单结果。其他数学领域的几个逻辑和组合方面已经被理解，甚至通过这种方法被发现。在这个领域有很多优秀的研究者，很多问题现在都是可以解决的

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.***