A quantitative bounded distance theorem and a Margulis' lemma for ℤn-actions, with applications to homology

We consider the stable norm associated to a discrete, torsionless abelian group of isometries Γ ≅ ℤnof a geodesic space (X, d). We show that the difference between the stable norm || ||stand the distance d is bounded by a constant only depending on the rank n and on upper bounds for the diameter of X̄ = Γ\X and the asymptotic volume ω(Γ, 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 Γ on (X, d); for this, we establish a lemma à la Margulis for ℤn-actions, which gives optimal estimates of ω(Γ, d) in terms of stsys(Γ, d), and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters n, diam(X̄) and ω(Γ, d) (or stsys(Γ, d)) are necessary to bound the difference d - || ||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 X̄ either is bounded by an explicit function of the first Betti number, diam(X̄) and ω(H1(X̄, ℤ)), or is a sublinear function of the mass.