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

Abstract

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 $\Vert {\Vert }_{st}$ 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 à 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 $d-\Vert {\Vert }_{st}$, 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.

14 pages. Upper bound of the Abelian Margulis' Lemma 1.2: Revised statement and argument of the characterization of the equality case. Corrected typos

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