Injectivity, cubical approximations and equivariant wall structures beyond CAT(0) cube complexes

This is an expository survey with two goals. 1) The primary goal is to discuss and highlight the impact of two recent influential ideas in geometric group theory. The first of which is the notion of an injective metric space which is a rich class of spaces that was imported to geometric group theory by Lang and have shown to be of a great effect. The second is Behrstock-Hagen-Sisto's cubical approximation theorem which provides a novel and particularly successful approach for studying mapping class groups (of finite type surfaces) and more generally, hierarchically hyperbolic groups. 2) Our second goal is to demonstrate how numerous geodesic metric spaces including hyperbolic spaces, CAT(0) spaces, and hierarchically hyperbolic spaces admit a strikingly rich equivariant wall structure: a discovery that was inspired by the aforementioned machines; the cubical approximation theorem as well as injective metric spaces.