Rigidity and Boundaries in Non-Positive Curvature

自从数学被研究以来，人们就一直试图理解几何与对称之间的关系。我们最熟悉的是欧几里得平面，紧随其后的是球面。人们早就知道，不能像对球体那样，用同样的形状排列来周期性地对平面进行平铺。这可以通过曲率的几何概念在数学上理解：平面是平的，而球体是正弯曲的。该项目涉及具有非正曲率的非欧几里得几何的广阔宇宙。PI将在这种情况下调查渐近不变量和刚性现象，同时通过指导和社区外展支持学生参与和扩大数学参与。本研究涉及有限生成群及其大尺度几何。第一个项目研究图形的离散性，这个概念结合了两个不同的研究项目：刚性现象和分类晶格包络。前者一直是几何群论的核心问题，而后者是由Mostow—Prasad刚性提出的，该刚性描述了双曲流形群的李群包络。PI将考虑各种各样的例子，包括具有门格尔曲线边界的双曲群和分裂为群图的群。第二个项目关注具有Menger紧实视觉边界的双曲群，并将建立技术来研究这些空间上的拟共形结构。第三个项目旨在通过解析方法和拟共形几何研究相对双曲群及其边界。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

As long as mathematics has been studied, people have sought to understand the relationship between geometry and symmetry. The Euclidean plane is most familiar, closely followed by the sphere. It has long been known that one cannot periodically tile the plane using the same arrangement of shapes as one would the sphere. This can be understood mathematically through the geometric notion of curvature: the plane is flat while the sphere is positively curved. This project concerns the vast universe of non-Euclidean geometries with non-positive curvature. The PI will investigate asymptotic invariants and rigidity phenomena in this setting, while supporting student involvement and broadened participation in mathematics via mentoring and community outreach. This research concerns finitely generated groups and their large-scale geometry. The first project investigates graphical discreteness, a notion that unifies two distinct programs of study: rigidity phenomena and classifying lattice envelopes. The former has been a central problem in geometric group theory, while the latter was initiated with Mostow--Prasad Rigidity, which characterized Lie group envelopes of hyperbolic manifold groups. The PI will consider a diverse family of examples, including hyperbolic groups with Menger curve boundary and groups that split as graphs of groups. The second project focuses on hyperbolic groups with Menger compacta visual boundaries and will build techniques to study the quasi-conformal structures on these spaces. The third project aims to study relatively hyperbolic groups and their boundaries via analytic methods and quasi-conformal geometry.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.