Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free product $\mathbb{Z}_{2} \star \mathbb{Z}_{2} = \langle a,b \mid a^{2} = e, b^{2} = e\rangle$?
I've tried doing something analogous to the case of the simply connected covering space of $RP^{2} \vee RP^{2}$ via even translation and the action of the antipodal map, but can't quite get it to work with this case.