Cyclic hyperbolicity in CAT(0) cube complexes

It is known that a cocompact special group G does not contain Z Z if and only if it is hyperbolic; and it does not contain F₂ Z if and only if it is toric relatively hyperbolic. Pursuing in this direction, we show that G does not contain F₂ F₂ if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT (0) cube complexes. Given such a group G, we first prove a structure theorem: G virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup H G either is virtually abelian or it admits a series H=H₀ H₁ Hₖ where Hₖ is acylindrically hyperbolic and where Hᵢ/H₈+₁ is finite or free abelian. As a consequence, G is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups.