Rigidity and Boundaries in Non-Positive Curvature

自从数学开始研究以来，人们一直在寻求理解几何和对称性之间的关系。欧几里德平面是最熟悉的，紧随其后的是球体。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平铺平面。这在数学上可以通过曲率的几何概念来理解：平面是平的，而球面是正弯曲的。这个项目关注的是具有非正曲率的非欧几里德几何的广阔宇宙。PI将调查渐近不变量和刚性现象在这种情况下，同时支持学生参与和扩大参与数学通过指导和社区推广。 本研究关注的是群生成及其大规模几何。第一个项目研究图形离散性，这一概念统一了两个不同的研究项目：刚性现象和分类格信封。前者一直是几何群论的中心问题，而后者则是由Mostow-Prasad刚性开始的，它刻画了双曲流形群的李群包络。PI将考虑不同的例子，包括具有门格尔曲线边界的双曲群和分裂为群的图的群。第二个项目的重点是双曲群与门格尔longta视觉边界，并将建立技术来研究这些空间上的准共形结构。第三个项目旨在通过分析方法和准共形几何研究相对双曲群及其边界。该奖项反映了NSF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而被认为值得支持。

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.