# Groups, Geometry, and Dynamics

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**Volume 10, Issue 4, 2016, pp. 1227–1247**

**DOI: 10.4171/GGD/381**

Published online: 2017-01-26

A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology

Filippo Cerocchi^{[1]}and Andrea Sambusetti

^{[2]}(1) Scuola Normale Superiore, Pisa, Italy

(2) Università di Roma La Sapienza, Italy

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 $\| \;\, \|_{\mathrm {st}}$ 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 $d -\| \;\, \|_{\mathrm {st}}$, 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.

*Keywords: *Systole, asymptotic volume, integral homology, stable norm, quasi-isometries

Cerocchi Filippo, Sambusetti Andrea: A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology. *Groups Geom. Dyn.* 10 (2016), 1227-1247. doi: 10.4171/GGD/381