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和对称性之间的关系。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，人们不能像对待球体那样周期性地平铺平面，使用相同的形状排列。这可以通过曲率的几何概念来理解：平面是平的，而球体是正曲线的。这个项目涉及具有非正曲率的非欧几里德几何的广阔宇宙。PI将研究这种环境下的渐近不变量和僵化现象，同时通过辅导和社区推广支持学生参与和扩大对数学的参与。这项研究涉及有限生成群及其大规模几何。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程序：刚性现象和分类格子包络。前者是几何群论中的一个中心问题，而后者是由Mostow-Prasad刚性引发的，它刻画了双曲流形群的李群包络。PI将考虑一系列不同的例子，包括具有Menger曲线边界的双曲群和分裂为群图的群。第二个项目集中于具有Menger紧视觉边界的双曲群，并将建立研究这些空间上的拟共形结构的技术。第三个项目旨在通过分析方法和准保角几何研究相对双曲群及其边界。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.