Abstract: Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mj (Mitra) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for normal hyperbolic subgroups of a hyperbolic group. These maps can be used to understand properties of such groups. I will explain why similar theorems fail for certain CAT(0) groups. This is joint work with Benjamin Beeker, Matthew Cordes, Giles Gardam, and Radhika Gupta.
Boundary maps in non-positive curvature