Geometric Group Theory

GERSTEN，98-00158 格斯滕计划继续他的工作同调不变量的几何群论。 他 给出了双曲群的同调和上同调特征以及群的子群的畸变的同调特征。 他计划使用这些不变量来研究有关格罗莫夫的有界上同调，字的问题，提出的群体，和同构问题的双曲群，等等。 几何群论领域使用几何学的方法来研究对称性。 这颠倒了F。世纪克莱因的埃尔朗格纲领，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其对称性来研究，其生成元是旋转、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，人们可以从欧几里德几何中恢复哪些对称性。 一个值得注意的事实是，文字问题，这涉及到编写一个计算机程序，以确定当两个对称性是相同的，当他们被写在发电机， 只取决于大尺度的几何形状。 文字问题是一个范例，为一大类问题，可以研究的方法，这一领域。

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.