JournalsggdVol. 10, No. 4pp. 1227–1247

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

  • Filippo Cerocchi

    Scuola Normale Superiore, Pisa, Italy
  • Andrea Sambusetti

    Università di Roma La Sapienza, Italy
A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology cover
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Abstract

We consider the stable norm associated to a discrete, torsionless abelian group of isometries ΓZn\Gamma \cong \mathbb Z^n of a geodesic space (X,d)(X,d). We show that the difference between the stable norm   st\| \;\, \|_{\mathrm {st}} and the distance dd is bounded by a constant only depending on the rank nn and on upper bounds for the diameter of Xˉ=Γ\X\bar X=\Gamma \backslash X and the asymptotic volume ω(Γ,d)\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 Γ\Gamma on (X,d)(X,d); for this, we establish a lemma à la Margulis for Zn\mathbb{Z}^n-actions, which gives optimal estimates of ω(Γ,d)\omega(\Gamma,d) in terms of stsys(Γ,d)(\Gamma,d), and vice versa, and characterize the cases of equality. Moreover, we show that all the parameters nn, diam(Xˉ)(\bar X) and ω(Γ,d)\omega (\Gamma, d) (or stsys (Γ,d)(\Gamma,d)) are necessary to bound the difference d  std -\| \;\, \|_{\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 Xˉ\bar X either is bounded by an explicit function of the first Betti number, diam(Xˉ)(\bar X) and ω(H1(Xˉ,Z))\omega(H_1(\bar X, \mathbb{Z})), 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 Zn\mathbb Z^n-actions, with applications to homology. Groups Geom. Dyn. 10 (2016), no. 4, pp. 1227–1247

DOI 10.4171/GGD/381