Quantitative Bounded Distance Theorem and Margulis' Lemma for Z^n actions with applications to homology

We consider the stable norm associated to a discrete, torsionless abelian group of isometries $\mathrm{\Gamma }\cong {\mathbb{Z}}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\mathrm{\parallel }{\mathrm{\parallel }}_{st}$ and the distance $d$ is bounded by a constant only depending on the rank $n$ and on upper bounds for the diameter of $\bar{X}=\mathrm{\Gamma }\mathrm{\backslash }X$ and the asymptotic volume $\omega (\mathrm{\Gamma },d)$. We also prove that the upper bound on the asymptotic volume is equivalent to a lower bound for the stable systole of the action of $\mathrm{\Gamma }$ on $(X,d)$; for this, we establish a Lemma `a la Margulis for ${\mathbb{Z}}^n$-actions, which gives optimal estimates of $\omega (\mathrm{\Gamma },d)$ in terms of $\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}(\mathrm{\Gamma },d)$, and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters $n,\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\bar{X})$ and $\omega (\mathrm{\Gamma },d)$ (or $$\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{y}\mathrm{s}$ (\Gamma,d)$)arenecessarytoboundthedifference$d -| ;, |_{st}$, by providing explicit counterexamples for each case. As an application, we prove that the number of connected components of any optimal, integral $1$-cycle in a closed Riemannian manifold $\bar{X}$ either is bounded by an explicit function of the first Betti number, $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\bar{X})$ and $\omega(H_1(\bar X, $\mathbb{Z}$), d)$, or is a sublinear function of the mass.