We prove a Margulis’ Lemma à la Besson-Courtois-Gallot, for manifolds whose fundamental group is a nontrivial free product , without 2-torsion. Moreover, if is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume-entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds.
Nous présentons une preuve d’un Lemme de Margulis à la Besson-Courtois-Gallot pour des variétés dont le groupe fondamental est un produit libre non trivial et sans élément de torsion d’ordre . De plus, quand est sans torsion nous donnons une minoration de la systole (homotopique) en fonction des bornes supérieurs sur le diamètre et sur l’entropie volumique. Nous allons fournir des exemples et des contre-exemples afin de montrer l’optimalité de nos hypothèses. Finalement, nous présentons deux applications de ce résultat : un théorème de précompacité et finitude et une estimation volumique pour variétés décomposables.
Keywords: Entropy, growth of groups, free products, systole
Mot clés : entropie, croissance des groupes, produits libres, systole
Cerocchi, Filippo 1
@article{AIF_2014__64_3_1011_0, author = {Cerocchi, Filippo}, title = {Margulis {Lemma,} entropy and free products}, journal = {Annales de l'Institut Fourier}, pages = {1011--1030}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {3}, year = {2014}, doi = {10.5802/aif.2872}, mrnumber = {3330162}, zbl = {06387299}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2872/} }
TY - JOUR AU - Cerocchi, Filippo TI - Margulis Lemma, entropy and free products JO - Annales de l'Institut Fourier PY - 2014 SP - 1011 EP - 1030 VL - 64 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2872/ DO - 10.5802/aif.2872 LA - en ID - AIF_2014__64_3_1011_0 ER -
%0 Journal Article %A Cerocchi, Filippo %T Margulis Lemma, entropy and free products %J Annales de l'Institut Fourier %D 2014 %P 1011-1030 %V 64 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2872/ %R 10.5802/aif.2872 %G en %F AIF_2014__64_3_1011_0
Cerocchi, Filippo. Margulis Lemma, entropy and free products. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1011-1030. doi : 10.5802/aif.2872. https://aif.centre-mersenne.org/articles/10.5802/aif.2872/
[1] Lipschitz precompactness for closed negatively curved manifolds, Proc. Amer. Math. Soc., Volume 127 (1999) no. 4, pp. 1201-1208 | MR | Zbl
[2] Finiteness theorems for Riemannian manifolds, Amer. J. Math., Volume 92 (1970), pp. 61-74 | MR | Zbl
[3] Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, 1639, Springer-Verlag, Berlin, 1996, pp. x+259 | MR | Zbl
[4] Hausdorff convergence of Riemannian manifolds and its applications, Recent topics in differential and analytic geometry (Adv. Stud. Pure Math.), Volume 18, Academic Press, Boston, MA, 1990, pp. 143-238 | MR | Zbl
[5] Un lemme de Margulis sans courbure et ses applications (2003) (Prépublication de l’Institut Fourier 595)
[6] Little topology, big volume, Duke Math. J., Volume 67 (1992) no. 2, pp. 273-290 | MR | Zbl
[7] Manifolds of negative curvature, J. Differential Geom., Volume 13 (1978) no. 2, pp. 223-230 | MR | Zbl
[8] Filling Riemannian manifolds, J. Differential Geom., Volume 18 (1983) no. 1, pp. 1-147 | MR | Zbl
[9] Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007, pp. xx+585 | MR | Zbl
[10] Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000, pp. vi+310 | MR | Zbl
[11] Correction to: “Gromov’s convergence theorem and its application” [Nagoya Math. J. 100 (1985), 11–48; MR0818156 (87e:53067)], Nagoya Math. J., Volume 114 (1989), pp. 173-174 | MR | Zbl
[12] Invariants topologiques et géométriques reliés aux longuers des géodésiques et aux sections harmoniques de fibrés, Université Grenoble 1, Institut Fourier (1994) (Ph. D. Thesis)
[13] Global and local volume bounds and the shortest geodesic loops, Comm. Anal. Geom., Volume 12 (2004) no. 5, pp. 1039-1053 | MR | Zbl
[14] Homological Group Theory, Lecture Notes, 36, London Mathematical Society, 1986
[15] Some finiteness results for groups with bounded algebraic entropy, Geom. Dedicata, Volume 143 (2009), pp. 49-62 | MR | Zbl
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