Minimal volume entropy and fiber growth

Ivan Babenko; Stéphane Sabourau

doi:10.5802/jep.295

Minimal volume entropy and fiber growth
[Entropie volumique minimale et croissance de fibres]

Ivan Babenko ¹ ; Stéphane Sabourau ²

¹ Université Montpellier II, CNRS UMR 5149, Institut Montpelliérain Alexander Grothendieck, Place Eugène Bataillon, Bât. 9, CC051, 34095 Montpellier Cedex 5, France
² Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France & Univ Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée, France

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 481-521.

Résumé
Abstract

Cet article traite d’hypothèses topologiques sous lesquelles l’entropie volumique minimale d’une variété fermée $M$, et plus généralement d’un complexe simplicial fini $X$, est nulle ou non nulle. Ces conditions topologiques sont exprimées en termes de croissance du groupe fondamental des fibres des applications d’un complexe simplicial fini $X$ vers des complexes simpliciaux $P$ de dimension inférieure. Cela conduit à une caractérisation complète des espaces ayant une entropie volumique minimale non nulle pour les complexes simpliciaux finis dont le groupe fondamental possède une croissance exponentielle uniforme sans sous-groupe de croissance intermédiaire. Comme nous l’a fait remarquer Vitali Kapovitch, ces conditions sont liées à l’effondrement avec courbure de Ricci minorée et permettent d’affiner le théorème d’isolement de Gromov. Nous donnons également des exemples de complexes simpliciaux finis ayant un volume simplicial nul et une entropie volumique minimale arbitrairement grande.

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold $M$, and more generally of a finite simplicial complex $X$, vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex $X$ to lower dimensional simplicial complexes $P$. This leads to a complete characterization of spaces with positive minimal volume entropy for finite simplicial complexes whose fundamental group has uniform exponential growth with no subgroup of intermediate growth. As pointed out to us by Vitali Kapovitch, these conditions are related to collapsing with Ricci curvature bounded below and lead to a refinement of Gromov’s isolation theorem. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.

Reçu le : 2024-04-24
Accepté le : 2025-03-20
Publié le : 2025-03-27

DOI : 10.5802/jep.295

Classification : 53C23, 57N65
Keywords: Minimal volume entropy, collapsing, exponential and subexponential growth, fiber growth, Urysohn width
Mots-clés : Entropie volumique minimale, effondrement, croissance exponentielle et sous-exponentielle, croissance des fibres, largeur d’Urysohn

Affiliations des auteurs :

Ivan Babenko ¹ ; Stéphane Sabourau ²

¹ Université Montpellier II, CNRS UMR 5149, Institut Montpelliérain Alexander Grothendieck, Place Eugène Bataillon, Bât. 9, CC051, 34095 Montpellier Cedex 5, France
² Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France & Univ Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Ivan Babenko; Stéphane Sabourau. Minimal volume entropy and fiber growth. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 481-521. doi : 10.5802/jep.295. https://jep.centre-mersenne.org/articles/10.5802/jep.295/

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