This three-year National Science Foundation Project Grant of $378,000 will support research into boundaries of groups in geometric group theory and hyperbolic geometry at the University of Illinois at Chicago from September 2022 through August 2025. Specifically, the principal investigator and collaborators will investigate the Cannon Conjecture regarding differences between classical and Gromov hyperbolic geometry in three dimensions in the presence of symmetries. They will continue developing theories of drilling hyperbolic groups with two-sphere boundaries and studying possible quasi-isometries between hyperbolic and relatively hyperbolic groups of this type. Additionally, the team will work to define coarse sectional curvature and apply it to coherence and local quasiconvexity of certain hyperbolic groups, as well as relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes. This award furthers the National Science Foundation's Mathematical and Physical Sciences program goal of advancing mathematical and physical sciences knowledge and understanding of major problems through support of meritorious research projects.