The alexandrov problem in a quotient space of ℍ²×ℝ

We prove an Alexandrov-type theorem for a quotient space of ℍ²×ℝ. More precisely, we classify the compact embedded surfaces with constant mean curvature in the quotient of ℍ²×R by a subgroup of isometries generated by a horizontal translation along horocycles of ℍ² and a vertical translation. We also construct some examples of periodic minimal surfaces in ℍ²×ℝ and we prove a multivalued Rado theorem for small perturbations of the helicoid in ℍ²×ℝ.