Mathematical Sciences: Topological and Geometric Aspects of Group Theory

给定一个群G和两个子群a，B，以及它们共有的另一个子群C，我们可以定义三元组（a，B；C）的“角”。这是自由产物与G的混合图；在合并自由积结构中，f的核的非平凡元素的最小长度是偶数2n。定义（A，B， C）角为pi/n。我们现在可以想象二维复合体K的每一个单元都有一个群，所以对于一个2单元的每一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是三角形时），我们可以定义一个“非球面”条件（如果K是三角形，这只是三个群论角度的和不超过pi的条件）；然后我们可以证明有一个包含2-配合物的所有标记基的大群H，并且这个群有一个K（H,1）配合物，这个配合物是以自然的方式构造的。这是通过拓扑结构证明的，但在这种情况下，有更严格的几何因素，这导致了对h群的更详细的了解。这个项目将继续沿着这个思路，推广到更复杂的2-复合体，尝试对作用于“非正曲率分段欧几里得2-复合体”的群进行分类，以格罗莫夫双曲群的精神将其推广到高维复合体，并将其与组合群论中的经典问题联系起来，看看这个理论会带来什么。群是具有乘法的代数系统，它满足一些自然规则。（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关注，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.