Geometric Group Theory

GERSTEN计划继续他在几何群论中同调不变量的研究。他给出了双曲群的同调和上同刻画，以及群的一个子群的畸变的同调刻画。他计划用这些不变量来研究关于Gromov的有界上同调、有限表示群的词问题、双曲群的同构问题等开放性问题。几何群论领域使用几何的方法来研究对称性。这颠覆了19世纪克莱因的厄兰格计划，该计划提出通过几何的对称性来研究几何。例如，在克莱因的计划中，欧几里得平面的几何将通过其对称性来研究，这些对称性的产生是旋转、反射和平行运动，而在几何群论中，人们将研究当人们忽略小尺度特征而只关注远处观察者可见的东西时，人们可以从欧几里得几何中恢复对称性的哪些特性。这是一个值得注意的事实，这个问题涉及到编写一个计算机程序来确定当两个对称用生成器表示时它们是否相同，它只依赖于大尺度几何。这个词问题是一大类问题的范例，这些问题可以用这个领域的方法来研究。

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.