What you wrote down is the quotient of the 2-dimensional translation group by a discrete subgroup. But by far not every closed manifold arises as a quotient of groups this way.

One should be aware that the term "compactification" in physics is used not so much to refer to what in mathematics is called [compactification](http://ncatlab.org/nlab/show/compactification) of non-compact spaces. For instace [one-point compactification](http://ncatlab.org/nlab/show/one-point%20compactification) in the mathematical sense turns the real line line into the circle. (However it also turns the plane into the 2-sphere, not into the torus.)

Instead, what is meant by "compactification" in physics is that you just _choose_ a [closed](http://ncatlab.org/nlab/show/closed%20manifold) (and hence compact) manifold $Q$, then choose spacetime $X$ to be a $Q$-fiber bundle over space base space (often assumed to be just a product $X = Q \times Y$), and then describe the [Kaluza-Klein mechanism](http://ncatlab.org/nlab/show/Kaluza-Klein%20mechanism) for passing from physics on $X = Q \times Y$ to effective physics on just $Y$.

In particular for Calabi-Yau "compactifications" you just _choose_ $Q$ to be a Calabi-Yau manifold, and then consider the Kaluza-Klein mechanism on spacetimes which are $Q$-fiber bundles. You don't actually obtain these spacetimes as compactifications of non-compact spacetimes in the sense of mathematics.

(Well one could consider that problem, but this is not what is generally meant by "compactification" in physics.)