A quantitative bounded distance theorem and a Margulis’ lemma for ${\mathbb{Z}}^n$-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 $\Vert {\Vert }_{\mathrm{st}}$ 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 stsys$(\Gamma ,d)$, and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters $n$, diam$(\bar{X})$ and $\omega (\Gamma ,d)$ (or stsys $(\Gamma ,d)$) are necessary to bound the difference $d-\Vert {\Vert }_{\mathrm{st}}$, by providing explicit counterexamples for each case.

As an application in Riemannian geometry, 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, diam$(\bar{X})$ and $\omega (H_1(\bar{X},\mathbb{Z}))$, or is a sublinear function of the mass.