Research in Geometry and Topology

DMS-0203045Mladen Bestvina 该项目涵盖几何、拓扑和几何群论的不同方面。统一的主题是具有大型对称群的空间的几何和拓扑研究。有时，首先出现的是群，例如代数对象的对称群，然后用反映该群的对称性构造空间。这允许在代数中使用几何/拓扑方法。该项目的另一个方面涉及寻找 3 维流形上的双曲结构。所有局部看起来像普通 3 维空间的（紧）空间实际上都是几何对象的可能性是由 W.P.瑟斯顿并在很多案例中得到验证。另一方面涉及群体上的丢番图几何学。这是对适用于特定类别群体的逻辑陈述的研究。令人惊讶的是，正如 Z. Sela 的工作所表明的那样，拓扑学进入了这项研究。更具体地说，该项目的主题是：特征簇的奇异性和特征簇的拓扑动力学、离散和忠实表示空间的紧性定理中的显式界限、弱双曲化猜想和3流形中的浸没表面叠层、低上同调维数的凯勒群的构造、群上的丢番图几何、托雷利群的有限性性质、 Coxeter、Artin 和相关群的刚性定理。

DMS-0203045Mladen BestvinaThe project encompasses different aspects of geometry, topology, and geometric group theory. The unifying theme is the geometric and topological study of spaces with large symmetry groups. Sometimes it is the group that arises first, for example as a symmetry group of an algebraic object, and then a space is constructed with symmetries reflecting this group. This allows for geometric/topological methods tobe used in algebra. Another aspect of the project involves finding hyperbolic structures on 3-dimensional manifolds. The possibility that all (compact) spaces that locally look like our ordinary 3-dimensional space are in fact geometric objects was first conceived by W.P. Thurston and verified in many cases. Yet another aspect involves Diophantinegeometry over groups. This is about a study of logical statements that hold for a particular class of groups. Surprisingly, as shown by the work of Z. Sela, topology enters this study; for example, surfaces appear naturally when one considers free groups.More specifically, the topics of the project are: Singularities of character varieties and topological dynamics on charactervarieties, Explicit bounds in the compactness theorems for spaces of discrete and faithful representations, Weak hyperbolization conjecture and immersed surface laminations in 3-manifolds, Constructions of Kaehler groups of low cohomological dimension, Diophantine geometry over groups,Finiteness properties of the Torelli group, Rigidity theorems for Coxeter, Artin, and related groups.