The class of grim reapers in ${\mathbb{H}}^2\times \mathbb{R}$

We study translators of the mean curvature flow in the product space \h^2\times\r. In \h^2\times\r there are three types of translations: vertical translations due to the factor \r and parabolic and hyperbolic translations from ${\text{h}}^2$. A grim reaper in \h^2\times\r is a translator invariant by a one-parameter group of translations. The variety of translators and translations in \h^2\times\r makes that the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of \h^2\times\r with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.