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 $\Gamma \cong \mathbb{Z}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\| \;\, \|_{st}$ and the distance $d$ is bounded by a constant only depending on the rank $n$ and on upper bounds for the diameter of $\bar X=\Gamma \backslash X$ and the asymptotic volume $\omega(\Gamma, d)$. 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 $\Gamma$ on $(X,d)$; for this, we establish a Lemma \`a la Margulis for $\mathbb{Z}^n$-actions, which gives optimal estimates of $\omega(\Gamma,d)$ in terms of $\mathrm{stsys}(\Gamma,d)$, and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters $n, \mathrm{diam}(\bar X)$ and $\omega (\Gamma, d)$ (or $\mathrm{stsys} (\Gamma,d)$) are necessary to bound the difference $d -\| \;\, \|_{st}$, 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{diam}(\bar X)$ and $\omega(H_1(\bar X, \mathbb{Z}), d)$, or is a sublinear function of the mass.