Dimensions of products of hyperbolic spaces

Estimates on asymptotic dimension are given for products of general hyperbolic spaces, with applications to hyperbolic groups. Examples are presented where strict inequality occurs in the product theorem for the asymptotic dimension in the class of hyperbolic groups and in the product theorem for the hyperbolic dimension. It is proved that $\mathbb{R}$ is dimensionally full for the asymptotic dimension in the class of hyperbolic groups.