Groups Acing on Hyperbolic Spaces and Operator Algebras

Geometric group theory studies groups by visualizing them as sets of transformations of metric spaces. This approach is particularly effective when the metric space satisfies certain negative curvature conditions, such as being hyperbolic. The proposed project builds upon the recent work of the PI in this direction. Specifically, the PI will make further advances in the study of groups acting on hyperbolic spaces and their operator algebras. Progressing toward the research objectives of this project requires expertise across various areas, including group theory, functional analysis, geometry, and dynamical systems. The PI will organize a series of week-long conferences aimed at fostering collaboration among experts and young researchers in these fields. Additionally, the project includes a range of educational activities targeting undergraduate and graduate students. The research project consists of three parts. The main goal of the first part is to study rigidity properties of the class of acylindrically hyperbolic groups. Driven by two major open problems regarding quasi-isometric and measure equivalence rigidity, the PI will address several auxiliary questions and conjectures that connect analytic and geometric properties of groups. In the second part, the PI will study groups acting cocompactly on simply connected, hyperbolic, simplicial complexes. Examples of groups admitting such actions with good control over the local data include hyperbolic and relatively hyperbolic groups, fundamental groups of graphs of groups and their small cancellation quotients, mapping class groups, many Artin groups, etc. The PI will generalize their previous results for relatively hyperbolic groups in this broader context. Lastly, the PI will continue work with collaborators on automorphisms of von Neumann algebras and reduced C*-algebras of countable groups. 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.