A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology
Filippo Cerocchi, Andrea Sambusetti · Groups Geometry and Dynamics · 2017
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.