Geometric Group Theory and 3-Manifolds

9803316 Stallings Stallings是工作的事项与Grushko的定理，与某些微妙的问题涉及的表面。 最近的一个版本涉及到一个映射的紧凑连接表面T到一个楔形的空间A和B，这是满射的基本群;如果边界curvesall映射到一边或另一边，除了可能有一个映射长度为2，映射是同伦的一个分裂的表面分为两部分沿着连接片。 他还感兴趣的事项concerninghyperbolic群体和削减顶点的图形和Heegaard分裂的3-流形。 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。 一个学生，金Whittlesey，解决了一个有趣的问题，关于存在的一个子群的映射类组，只包括平凡元素连同伪Anosov地图。 另一个学生正在研究关于自由群的算法问题，并正在深入研究自由群中的方程理论。 另一个学生对使用计算机来发现几何物体感兴趣，另一个学生已经开始深入研究自动群和双曲群的一些细节。 补助金的主要部分是部分资助学生。 群论最初是研究几何或代数对象的对称性。 几何学可以软化为拓扑学，这产生了一些有趣的新想法;格罗莫夫是这个（20世纪80年代）的创始人之一，自从他的第一部作品以来，它已经大大扩展了。 事后看来，人们可以看到，关于群的几何思想出现在高斯（约1800年）、庞加莱（约1900年）和怀特海（1930年）的工作中。三维流形的性质尤其受到基本群的影响，相反，许多关于群的结果也受到这些拓扑思想的启发。 其他数学领域的一些逻辑和组合方面已经被这种方法所理解甚至发现。 在这个领域有许多优秀的研究人员，许多问题现在是可以解决的。

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.***