Mathematical Sciences: Topological and Geometric Aspects of Group Theory

8905777给定一个组G和两个子组A、B，再加上两个共同的子组C，我们就可以定义三和弦(A，B；C)的“角度”。存在合并的自由积到G中的映射f；就合并的自由积结构而言，f的核的非平凡元素的最小长度是偶数2n。(A，B；C)的角度被定义为pi/n。我们现在可以想象，对于二维复数K的每个单元，都有一个关联的群，使得对于一个2-单元的每个角，都有一个嵌入在顶点群中的三元组。现在，在某些情况下(例如，当K是一个三角形时)，我们可以定义一个“非球形”条件(如果K是一个三角形，这仅仅是三个群论角度之和至多为pi的条件)；然后我们可以证明存在一个包含2-复形的所有标号群的大群H，并且这个群有一个以自然方式构造的K(H，1)复形。这一点被一个拓扑结构证明了，但在这种情况下有更严格的几何事物，这导致了对群H的更详细的理解。这项工作是最近的。该项目将继续沿用这一思路，将其推广到更复杂的2-复形，试图对作用于“非正曲率的分段欧几里德2-复形”的群进行分类，按照格罗莫夫双曲群的精神将其推广到更高维的复形，将其与组合群论中的经典问题联系起来，并看看这一理论将引向何方。群是具有满足一些自然规则的乘法的代数系统。(这些规则是从对称组中抽象出来的。)这个概念在数学和理论物理中无处不在，群论的任何实质性进展都可能受到大量听众的追随，并产生广泛的反响。

8905777 Stallings Given a group G, and two subgroups A, B, together with a further subgroup C common to both, we can define the "angle" of the triad (A,B;C). There is a map f of the amalgamated free product into G; the smallest length of a non-trivial element of the kernal of f, in terms of the amalgamated free product structure, is an even number 2n. The angle of (A,B;C) is defined to be pi/n. We can now imagine that to each cell of a two-dimensional complex K there is associated a group, so that to each corner of a 2-cell there is associated a triad embedded in the vertex group. Now, in some cases (for instance, when K is a triangle), we can define a "non-spherical" condition (if K is a triangle, this is simply the condition that the sum of the three group-theoretic angles is at most pi); and we can then prove that there is a big group H containing all the label groups of the 2-complex, and that this group has a K(H,1) complex which is constructed in a natural way. This is proved by a topological construction, but there are more rigidly geometrical things underlying the situation, which lead to a more detailed understanding of the group H. This work is very recent. The project is to continue in this vein, to generalize to more complicated 2-complexes, to try to classify groups acting on "piecewise-Euclidean 2-complexes of non-positive curvature," to generalize to higher-dimensional complexes in the spirit of Gromov's hyperbolic groups, to relate this to classical problems in combinatorical group theory, and to see where this theory leads. Groups are algebraic systems with a multiplication which satisfies a few natural rules. (These rules are abstracted from groups of symmetries.) The concept is ubiquitous in mathematics and theoretical physics, and any substantial advances in group theory are likely to be followed by a large audience and to have wide repercussions.