Workshop on Nonpositively Curved Groups

非正曲群研讨会

• 批准号: 1822310
• 负责人: Kim Ruane
• 金额: $ 2.8万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-05-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1822310&HistoricalAwards=false
• 关键词: Workshop; Nonpositively; Curved; Groups

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.

关于非正弯曲群的研讨会将于2018年5月23日至29日在以色列的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）和丹尼尔怀斯（麦吉尔大学）合作组织。 无限群出现在拓扑空间和几何学的研究中。 拓扑空间的基本群是一个与空间相关的代数对象，它本质上描述了任何“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--这些也形成一个群。 在代数拓扑学和几何群论中，有一个信息交换，即代数通知拓扑或几何，反之亦然。 在数学中，人们经常试图理解一个特定类型的所有对象如下：首先理解一些基本的具体例子，然后试图表明这些对象中的一个通用对象要么与基本例子相同，要么只是以一种容易描述的方式与之不同。 在双曲三维流形的背景下，哈肯流形是那些包含一个2边不可压缩曲面的流形。 在1960年代，哈肯表明，如果这样的表面是存在的，那么原始的3-流形可以完全描述粘合在一起的许多加厚的表面以一种特殊的方式。 在Perelman证明了几何化猜想之后，三维流形拓扑学中最大的问题之一就是虚哈肯猜想。 这基本上断言任何紧致双曲三维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三维流形的基本群几乎可以被实现为一个非正曲立方复形的对称群。 这是一个定理在几何群论有重大后果，在几何，拓扑和群论。 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。 研讨会的主题是代数，几何和分析方面的团体，作用于CAT（0）空间的等距。CAT（0）空间是Gromov在20世纪80年代引入的，作为非正截面曲率的黎曼流形的推广，包含了丰富的一类不是流形的度量空间。 这些空间的经典例子包括非紧型的对称空间、欧几里得和双曲建筑，以及它们的有限笛卡尔积。许多例子承认适当的群作用，这些群在经典情况下通常是算术的。单纯树是CAT（0）空间最简单的例子，作用于树的Bass-Serre群论是几何群论的重要早期章节。CAT（0）立方体复形是单纯树的自然高维推广。 这些立方体复形现在因其在上述双曲三维流形的虚哈肯猜想的最近解决方案中的核心作用而闻名。 研讨会的目的是汇集初级和高级人员在该地区工作，讨论进一步开放的问题，这些度量空间，以及竞争形式的非度量组合nonpositive curviline.Workshop网站：http://cms-math.net.technion.ac.il/nonpositively-curved-groups-on-the-mediterranean/This奖反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。