I've just been introduced to the idea of geometric dimension of a group (smallest dimension of $K(G,1)$) and I'm trying to figure out what the geometric dimension of $\mathbb{Z}^n$ is for $n\geq 3$. For $\mathbb{Z}^1$ and $\mathbb{Z}^2$ I've figured them out, with some helpful results about $gd(G)=1$ groups.
For what it is worth, I know that $gd(\mathbb{Z}^n)\leq n$ since $T^n$ is a $K(\mathbb{Z}^n,1)$ for every $n$... since each one has universal cover $\mathbb{R}^n$. Other than that, I am a bit stumped about how to move forward. If there's a catalogue, that would be even better. Another suggestion for an answer: help with thinking about $\mathbb{Z}^3$.
If $gd(\Gamma) < n$ then $H_n(K(\Gamma,1);\mathbb Z)$ is trivial, because there is a $K(\Gamma,1)$ with no cells in dimension $n$ (by definition of $gd$ and using that $gd(\Gamma)<n$).
Also, $H_n(K(\Gamma,1);\mathbb Z)$ is independent of the choice of $K(\Gamma,1)$, because $K(\Gamma,1)$ is well-defined up to homotopy equivalence.
So, if $gd(\mathbb Z^n)$ were $<n$ then $H_n(K(\mathbb Z^n,1);\mathbb Z)$ would be trivial. But it's not: as you say, $T^n$ is a $K(\mathbb Z^n,1)$, and $H_n(T^n;\mathbb Z)=\mathbb Z$.