  It is known that a cocompact special group $G$ does not contain $\mathbb{Z}
\times \mathbb{Z}$ if and only if it is hyperbolic; and it does not contain
$\mathbb{F}_2 \times \mathbb{Z}$ if and only if it is toric relatively
hyperbolic. Pursuing in this direction, we show that $G$ does not contain
$\mathbb{F}_2 \times \mathbb{F}_2$ 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 $H \leq G$ either is virtually abelian or it admits a series $H=H_0
\rhd H_1 \rhd \cdots \rhd H_k$ where $H_k$ is acylindrically hyperbolic and
where $H_i/H_{i+1}$ 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.
