Boundaries of Groups

几个世纪以来，数学的一个中心主题一直是代数和几何之间的相互作用。 通常的方向是研究感兴趣的几何对象的对称性。 在几何群论中，这成为一条双向的道路，因为代数对象（如群）被认为是几何对象。双曲几何是一个主题回到工作的波尔约，高斯和其他人在世纪，但它也发挥了核心作用，在现代几何，由于影响瑟斯顿和格罗莫夫。 这个项目围绕着几何群论中的一个中心问题，坎农猜想，关于经典双曲几何和格罗莫夫的粗糙概念之间的差异（在三维空间中），在存在一大群对称性的情况下。更广泛的影响，这一项目包括研究培训的机会，为研究生。在过去的十年中，首席研究员，沿着与曼宁和其他人，已经开发了许多工具，涉及相对双曲德恩填充，这给了强有力的控制某些种类的相对双曲群的导数。 这个项目利用这项工作来研究双曲和相对双曲群，其边界在无穷远处是一个双球面。 坎农猜想预言这样的群实际上是克莱因群。 该项目提出了各种方法，这和相关的架构。 与海辛斯基，曼宁，Osajda，西斯托和沃尔什，PI继续发展理论的钻探双曲群与两个球边界。 PI将研究双曲群和相对双曲群之间可能的准等距性。 与威尔顿，PI将开发一个概念的粗截面曲率，与应用程序的连贯性和局部准凸的某些双曲群。 在另一个不同但相关的方向上，PI将与爱因斯坦一起继续研究CAT（0）立方体复合体上相对双曲群的相对几何作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

A central theme in mathematics for centuries has been the interaction between algebra and geometry. The usual direction is to study the set of symmetries of a geometric object of interest. In geometric group theory, this becomes a two-way street in that algebraic objects (such as groups) are considered as geometric objects in their own right. Hyperbolic geometry is a subject going back to work of Bolyai, Gauss and others in the 19th Century, but it also plays a central role in modern geometry, due to the influence of Thurston and Gromov. This project centers around a central question in geometric group theory, the Cannon Conjecture, about the difference (in three dimensions) between classical hyperbolic geometry and the coarse notion due to Gromov, in the presence of a large group of symmetries. Broader impacts of this project include research training opportunities for graduate students.Over the last decade the principal investigator, along with Manning and others, has developed many tools involving relatively hyperbolic Dehn filling, which gives strong control on certain kinds of quotients of relatively hyperbolic groups. This project leverages this work to study hyperbolic and relatively hyperbolic groups whose boundary at infinity is a two-sphere. The Cannon Conjecture predicts that such groups are virtually Kleinian groups. This project proposes various approaches to this and related conjectures. With Haissinsky, Manning, Osajda, Sisto and Walsh, the PI continues to develop a theory of drilling hyperbolic groups with two-sphere boundary. The PI will investigate possible quasi-isometries between hyperbolic and relatively hyperbolic groups with 2-sphere boundaries. With Wilton, the PI will develop a notion of coarse sectional curvature, with applications to coherence and local quasi-convexity of certain hyperbolic groups. In a different but related direction, with Einstein the PI will continue to study relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes.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.