Geodesic
A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology
Filippo Cerocchi1 ; Andrea Sambusetti2
1Scuola Normale Superiore, Pisa, Italy
2Università di Roma La Sapienza, Italy
Groups, geometry, and dynamics, Tome 10 (2016) no. 4, pp. 1227-1247

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DOI

We consider the stable norm associated to a discrete, torsionless abelian group of isometries Γ≅Zn of a geodesic space (X,d). We show that the difference between the stable norm ∥∥st​ and the distance d is bounded by a constant only depending on the rank n and on upper bounds for the diameter of Xˉ=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(Xˉ) and ω(Γ,d) (or stsys (Γ,d)) are necessary to bound the difference d−∥∥st​, by providing explicit counterexamples for each case.
DOI : 10.4171/ggd/381
Classification : 53-XX, 52-XX, 57-XX
Mots-clés : Systole, asymptotic volume, integral homology, stable norm, quasi-isometries

Filippo Cerocchi  1   ; Andrea Sambusetti  2

1 Scuola Normale Superiore, Pisa, Italy
2 Università di Roma La Sapienza, Italy 
Filippo Cerocchi; Andrea Sambusetti. A quantitative bounded distance theorem and a Margulis’ lemma for $\mathbb Z^n$-actions, with applications to homology. Groups, geometry, and dynamics, Tome 10 (2016) no. 4, pp. 1227-1247. doi: 10.4171/ggd/381
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     author = {Filippo Cerocchi and Andrea Sambusetti},
     title = {A quantitative bounded distance theorem and a {Margulis{\textquoteright}} lemma for $\mathbb Z^n$-actions, with applications to homology},
     journal = {Groups, geometry, and dynamics},
     pages = {1227--1247},
     year = {2016},
     volume = {10},
     number = {4},
     doi = {10.4171/ggd/381},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/381/}
}
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