  We prove an Alexandrov type theorem for a quotient space of $\mathbb
H^2\times \mathbb R$. More precisely we classify the compact embedded surfaces
with constant mean curvature in the quotient of $\mathbb H^2\times \mathbb R$
by a subgroup of isometries generated by a parabolic translation along
horocycles of $\mathbb H^2$ and a vertical translation. Moreover, we construct
some examples of periodic minimal surfaces in $\mathbb H^2\times\mathbb R$ and
we prove a multi-valued Rado theorem for small perturbations of the helicoid in
$\mathbb H^2\times\mathbb R$.
