We consider the stable norm associated to a discrete, torsionless abelian group of isometries of a geodesic space . We show that the difference between the stable norm and the distance is bounded by a constant only depending on the rank and on upper bounds for the diameter of and the asymptotic volume . 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 on ; for this, we establish a lemma à la Margulis for -actions, which gives optimal estimates of in terms of stsys, and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters , diam and (or stsys ) are necessary to bound the difference , 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 either is bounded by an explicit function of the first Betti number, diam and , or is a sublinear function of the mass.
Cite this article
Filippo Cerocchi, Andrea Sambusetti, A quantitative bounded distance theorem and a Margulis’ lemma for -actions, with applications to homology. Groups Geom. Dyn. 10 (2016), no. 4, pp. 1227–1247DOI 10.4171/GGD/381