Workshop on Nonpositively Curved Groups

2018年5月23日至29日，一场关于非正弯曲群的研讨会将在以色列的Nachsholim举行。研讨会将由Kim Ruane（塔夫茨大学）与Michah Sageev（以色列理工学院）和Daniel Wise（麦吉尔大学）合作组织。无穷群出现在拓扑空间和几何的研究中。拓扑空间的基本群是与空间相关的代数对象，它本质上描述了存在的任何“洞”的数量和结构。在几何学中，我们可以研究空间的对称性或刚性运动——这些也构成了一个群体。在代数拓扑和几何群论中，存在一种信息交换，即代数告知拓扑或几何，反之亦然。在数学中，人们通常试图这样理解某一特定类型的所有对象：首先理解一些基本的具体例子，然后试图证明这些对象中的一个一般对象要么与一个基本例子相同，要么只是在一种容易描述的方式上与它不同。在双曲型3-流形的情况下，Haken流形是包含2面不可压缩曲面的流形。在20世纪60年代，Haken证明了如果存在这样的曲面，那么原始的3流形可以通过以特定的方式将有限多个加厚的曲面粘合在一起来完全描述。在佩雷尔曼证明了几何化猜想之后，三流形拓扑中遗留下来的最大问题之一就是虚哈肯猜想。这基本上断言任何紧致双曲3流形要么是哈肯流形，要么与哈肯流形密切相关。最近，伊恩·阿戈尔（Ian Agol）的获奖作品证明了这一点。这个难题的一个关键部分是证明任何双曲3流形的基本群几乎可以被实现为一个非正弯曲立方体复合体的对称群。这是几何群论中的一个定理，在几何、拓扑学和群论中都有重要的意义。非正曲率的作用不能被夸大，因此我们的研讨会旨在进一步探索这种联系。研讨会的主题是通过等距作用于CAT(0)空间的群的代数、几何和分析方面。CAT(0)空间是Gromov在20世纪80年代作为非正截面曲率黎曼流形的推广引入的，它包含了一类丰富的非流形度量空间。这些空间的经典例子包括非紧凑型对称空间，欧几里得和双曲建筑，以及它们的有限笛卡尔积。许多例子都承认适当的群体行为，而这些群体在经典情况下通常是算术的。简单树是CAT(0)空间中最简单的例子，而作用于树的群的Bass-Serre理论是几何群论的重要早期章节。CAT(0)立方复合体是简单树的自然高维推广。这些立方体复合体现在因其在前面提到的双曲3-流形的虚哈肯猜想的最新解中所起的中心作用而闻名。研讨会旨在将该领域的初级和高级人员聚集在一起，讨论有关这些度量空间以及非度量组合非正曲率的竞争形式的进一步开放问题。该奖项反映了美国国家科学基金会的法定使命，并通过基金会的知识价值和更广泛的影响审查标准进行评估，认为值得支持。

A workshop on Nonpositively Curved Groups will take place at Nachsholim in Israel from May 23-29, 2018. The workshop will be organized by Kim Ruane (Tufts University) in collaboration with Michah Sageev (Technion) and Daniel Wise (McGill University). Infinite groups arise in the study of topological spaces and in geometry. The fundamental group of a topological space is an algebraic object associated to the space which essentially describes the number and structure of any "holes" that are there. In geometry, we can study the symmetries or rigid motions of the space - these also form a group. In Algebraic Topology and in Geometric Group Theory, there is an exchange of information whereby the algebra informs the topology or geometry and vice versa. In mathematics, one often attempts to understand all objects of a particular type as follows: first understand some fundamental concrete examples and then try to show that a generic one of these objects is either the same as one of the fundamental examples or only differs from it in a way that can be easily described. In the setting of hyperbolic 3-manifolds, the Haken manifolds are those that contain a 2-sided incompressible surface. In the 1960's, Haken showed that if such a surface was there, then the original 3-manifold can be completely described by gluing together finitely many thickened up surfaces in a particular way. One of the biggest problems left open in 3-manifold topology after the Geometrization Conjecture was proved by Perelman was the Virtual Haken Conjecture. This basically asserts that any compact hyperbolic 3-manifold is either Haken or is closely related to one that is Haken. This was recently shown to be true by the award winning work of Ian Agol. A key piece of the puzzle was to show that the fundamental group of any hyperbolic 3-manifold can almost be realized as a group of symmetries of a nonpositively curved cube complex. This is a theorem in Geometric Group Theory which has significant consequences in geometry, topology and in group theory. The role of nonpositive curvature cannot be overstated and so our workshop aims to explore this connection further. The theme of the workshop is algebraic, geometric and analytical aspects of groups that act on CAT(0) spaces by isometries. CAT(0) spaces were introduced by Gromov in the 1980's as a generalization of Riemannian manifolds of nonpositive sectional curvature and encompass a rich class of metric spaces that are not manifolds. Classical examples of these spaces include symmetric spaces of non-compact type, Euclidean and hyperbolic buildings, as well as finite Cartesian products of these. Many examples admit proper group actions, and these groups are often arithmetic in the classical case. Simplicial trees are the simplest examples of CAT(0) spaces and the Bass-Serre theory of groups acting on trees is an important early chapter of geometric group theory. CAT(0) cube complexes are a natural high dimensional generalization of simplicial trees. These cube complexes are now famous for their central role in the recent solution of the Virtual Haken Conjecture for hyperbolic 3-manifolds mentioned above. The workshop aims to bring together junior and senior people working in the area to discuss further open problem concerning these metric spaces as well as competing forms of non-metric combinatorial nonpositive curvature.Workshop Website: http://cms-math.net.technion.ac.il/nonpositively-curved-groups-on-the-mediterranean/This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.