$SL_2(\Bbb Z)$ acts on ${\Bbb R}^2$ fixing set-wise ${\Bbb Z}^2$, so $SL_2(\Bbb Z)$ acts on ${\Bbb R}^2\setminus {\Bbb Z}^2$, and then on the universal covering space of ${\Bbb R}^2\setminus {\Bbb Z}^2$, call it $U$.
Can one not then give $U$ a hyperbolic metric so that $SL_2(\Bbb Z)$ acts by isometries?
And doing so, does one recover, up to conjugacy, the usual action of $SL_2(\Bbb Z)$ on the hyperbolic disk?
Finally then, how explicit can one make the uniformization of $U$?
Edit: Mischa's spot-on comments make this a question in search of a better question. I considered deleting, but perhaps some further interesting comments will emerge.