There is a sense in which $\mathbb{Z}$ "looks like" a line and $\mathbb{Z}_n$ looks like a circle.
For two dimensions / two generators, $\mathbb{Z}\times\mathbb{Z}$ looks like a plane, $\mathbb{Z}\times\mathbb{Z}_n$ like a cylinder, and $\mathbb{Z}_m\times\mathbb{Z}_n$ like a torus.
Is there a way to more formally define this intuitive correspondence? Preferably in such a way that some easily definable subset of $n$-manifolds is one-to-one with some easily definable class of algebraic structures on $n$ generators?