Vertex numbers of simplicial complexes with free abelian fundamental group

We show that the minimum number of vertices of a simplicial complex with fundamental group ${\mathbb{Z}}^n$ is at most $O(n)$ and at least $\mathrm{\Omega }(n^{3\mathrm{/}4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$. For the lower bound, we use a fractional Sylvester-Gallai result. We also prove that any group presentation $\langle S | R\rangle \cong $\mathbb{Z}$^{n}$whoserelationsareoftheform$g^{a}h^{b}i^{c}$for$g, h, i \in S$hasatleast$\Omega(n^{3/2})$ generators.