Relatively geometric actions on CAT$\operatorname{CAT}$(0) cube complexes

We develop the foundations of theory relatively geometric actions hyperbolic groups on CAT(0) cube complexes, a notion introduced in our previous work [5]. In setting we prove: full quasi-convex subgroups are convex compact; an analog Agol's Theorem; and version Haglund--Wise's Canonical Completion Retraction.