Infinite Discrete Groups

这个项目有两条线索：群的代数不变量与几何不变量，以及探索非正曲线群。这两个都属于格罗莫夫通过大规模几何学研究无限离散群的轨道，这在很大程度上是由与黎曼几何学的平行关系驱动的。第一个重点是填充不变量，它涉及与群相关的空间中跨越环的圆盘的几何特征(“肥皂膜几何”)。第二个问题是关于群论中的非正曲率概念与非正曲群中存在的野性之间的相互关系，特别是在它们的共轭问题和子群中。这项工作属于几何群论的主题，它涉及无限群及其作用的空间。它起源于20世纪初，特别是低维和代数拓扑学。它从20世纪80年代末开始发展起来，当时人们清楚地看到，所涉及空间的几何特征对群的代数结构有深远的影响。丰富的相互作用随之而来，例如，黎曼几何、概率、李群、动力系统、遍历理论、组合学、计算机科学和逻辑。除了研究工作，几何和拓扑学课程将被设计用于学习不是主要集中在数学上的本科生。

This project has two strands: algebraic versus geometric invariants of groups, and exploring non-positively curved groups. Both fall within the orbit of Gromov's study of infinite discrete groups through large-scale geometry, heavily driven by parallels with Riemannian geometry. The first focuses on filling invariants, which concern the geometric features of discs spanning loops ("soap film geometry") in spaces associated with groups. The second concerns how notions of non-positive curvature in group theory interrelate and the wildness present in non?positively curved groups, particularly in their conjugacy problems and subgroups.The work falls within the subject of Geometric Group Theory, which concerns infinite groups and the spaces on which they act. Its roots lie in the early 20th century, particularly in low-dimensional and algebraic topology. It has taken off since the late 1980s when it became clear that geometric features of the spaces involved have profound repercussions for the algebraic structure of the groups. Rich interactions have ensued with, for example, Riemannian geometry, probability, Lie groups, dynamical systems, ergodic theory, combinatorics, computer science and logic. In addition to the research work, a course in Geometry and Topology will be deigned for undergraduates whose studies who are not primarily concentrated in mathematics.