Geometric Group Theory

格尔斯滕，98-00158格尔斯滕计划继续他在几何群论中同调不变量的工作。他给出了双曲群的同调和上同调刻画以及群的子群扭曲的同调刻画。他计划使用这些不变量来研究关于Gromov的有界上同调、有限表示群的字问题和双曲群的同构问题等公开问题。几何群论领域使用几何方法来研究对称性。这颠覆了19世纪F·克莱因的Erlanger计划，该计划提出通过几何的对称性来研究几何。例如，在克莱因的程序中，欧几里德平面的几何将通过它的对称性来研究，它的生成器是旋转、反射和平行运动，而在几何群论中，当人们忽略小尺度特征而只关注远处观察者所能看到的东西时，人们将研究人们可以从欧几里德几何中恢复出对称的哪些性质。一个值得注意的事实是，单词问题只取决于大规模的几何。问题涉及编写计算机程序，以确定当两个对称以生成器的形式编写时，它们何时相同。问题一词是一大类问题的范例，可以用这一领域的方法来研究。

GERSTEN, 98-00158 Gersten plans to continue his work on homological invariants in geometric group theory. He has given both homological and cohomological characterizations of hyperbolic groups and a homological characterization of distortion of a subgroup of a group. He plans to use these invariants to study open questions concerning Gromov's bounded cohomology, the word problem for finitely presented groups, and the isomorphism problem for hyperbolic groups, among others. The area of geometric group theory uses the methods of geometry to study symmetry. This turns upside down F. Klein's Erlanger Program of the 19th century, which proposed studying a geometry by means of its symmetries. For example, in Klein's program the geometry of the Euclidean plane would be studied by means of its symmetries, whose generators are rotations, reflections and parallel motions, whereas in geometric group theory, one would study which properties of the symmetries one could recover from Euclidean geometry, when one ignores small-scale features and concentrates only on what is visible to a distant observer. It is a remarkable fact that the word problem, which concerns writing a computer program to decide when two symmetries are the same when they are written in terms of generators, depends only on the large-scale geometry. The word problem is a paradigm for a large class of problems which can be studied by the methods of this area.