Nous initions une étude quantitative de l’équivalence mesurée entre groupes de type fini. Ce faisant nous élargissons le contexte devenu classique de l’équivalence mesurée . Dans cet article, nous nous concentrons sur le cas des groupes moyennables, classe pour laquelle nous démontrons un théorème de rigidité ainsi que plusieurs résultats de flexibilité.
Notre résultat de rigidité stipule que le profil isopérimétrique possède une propriété de monotonicité très générale, impliquant en particulier son invariance sous équivalence mesurée intégrable. Ceci fournit des « bornes inférieures » explicites sur le degré d’intégrabilité des couplages mesurés possibles entre deux groupes moyennables. Ce résultat a également une application inattendue en géométrie des groupes : le profil isopérimétrique est monotone sous plongement grossier entre groupes moyennables. Entre autres applications, cela entraine l’existence d’un continuum de groupes -résolubles n’admettant aucun plongement grossier les uns dans les autres.
Nos résultats de flexibilité consistent à exhiber des équivalences orbitales explicites aux propriétés d’intégrabilité prescrites entre certains groupes moyennables. Nous introduisons pour cela un nouvel outil : les suites de pavages de Følner. Nous en déduisons dans de nombreux cas que l’obstruction quantitative obtenue à l’aide du profil isopérimétrique est optimale à un facteur d’erreur logarithmique près. Nous obtenons en outre que deux invariant importants de quasi-isométrie ne sont pas préservés sous équivalence orbitale intégrable : la dimension asymptotique, et le fait d’être de présentation finie.
We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results.
On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under measure equivalence. This yields explicit “lower bounds” on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of -solvable groups which pairwise do not coarsely embed into one another.
On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of Følner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under orbit equivalence: the asymptotic dimension and finite presentability.
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@article{AHL_2022__5__1417_0, author = {Delabie, Thiebout and Koivisto, Juhani and Le Ma{\^\i}tre, Fran\c{c}ois and Tessera, Romain}, title = {Quantitative measure equivalence between amenable groups}, journal = {Annales Henri Lebesgue}, pages = {1417--1487}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.155}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ahl.155/} }
TY - JOUR AU - Delabie, Thiebout AU - Koivisto, Juhani AU - Le Maître, François AU - Tessera, Romain TI - Quantitative measure equivalence between amenable groups JO - Annales Henri Lebesgue PY - 2022 SP - 1417 EP - 1487 VL - 5 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.155/ DO - 10.5802/ahl.155 LA - en ID - AHL_2022__5__1417_0 ER -
%0 Journal Article %A Delabie, Thiebout %A Koivisto, Juhani %A Le Maître, François %A Tessera, Romain %T Quantitative measure equivalence between amenable groups %J Annales Henri Lebesgue %D 2022 %P 1417-1487 %V 5 %I ÉNS Rennes %U http://www.numdam.org/articles/10.5802/ahl.155/ %R 10.5802/ahl.155 %G en %F AHL_2022__5__1417_0
Delabie, Thiebout; Koivisto, Juhani; Le Maître, François; Tessera, Romain. Quantitative measure equivalence between amenable groups. Annales Henri Lebesgue, Tome 5 (2022), pp. 1417-1487. doi : 10.5802/ahl.155. http://www.numdam.org/articles/10.5802/ahl.155/
[Aus16] Integrable Measure Equivalence for Groups of Polynomial Growth, Groups Geom. Dyn., Volume 10 (2016) no. 1, pp. 117-154 | DOI | MR | Zbl
[Bau61] Wreath Products and Finitely Presented Groups, Math. Z., Volume 75 (1961) no. 1, pp. 22-28 | DOI | MR | Zbl
[BD08] Asymptotic Dimension, Topology Appl., Volume 155 (2008) no. 12, pp. 1265-1296 | DOI | MR | Zbl
[Ben12] Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth, J. Theor. Probab., Volume 4 (2012), pp. 631-637 | DOI | MR | Zbl
[BFS13] Integrable Measure Equivalence and Rigidity of Hyperbolic Lattices, Invent. Math., Volume 194 (2013) no. 2, pp. 313-379 | DOI | MR | Zbl
[Bri14] Growth Behaviors in the Range , Afr. Mat., Volume 25 (2014) no. 4, pp. 1143-1163 | DOI | MR | Zbl
[BST12] On the Separation Profile of Infinite Graphs, Groups Geom. Dyn., Volume 6 (2012) no. 4, pp. 639-658 | DOI | MR | Zbl
[BZ21] Speed of Random Walks, Isoperimetry and Compression of Finitely Generated Groups, Ann. Math., Volume 193 (2021) no. 1, pp. 1-105 | DOI | MR | Zbl
[CGP01] A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier, Volume 51 (2001) no. 6, pp. 1763-1827 | DOI | MR | Zbl
[Cou00] Random Walks and Geometry on Infinite Graphs, Lectures Notes on Analysis in Metric Spaces (Trento, 1999) (Ambrosio, Luigi; Serra Cassano, Francesco, eds.) (Appunti Corsi Tenuti Docenti Sc.), Scuola Normale Superiore di Pisa (2000), pp. 5-36 | Zbl
[CS93] Isopérimétrie Pour Les Groupes et Les Variétés, Rev. Mat. Iberoam., Volume 9 (1993) no. 2, pp. 293-314 | DOI | Zbl
[de08] Dimension of asymptotic cones of Lie groups, J. Topol., Volume 1 (2008) no. 2, pp. 342-361 | DOI | MR | Zbl
[DS06] Asymptotic Dimension of Discrete Groups, Fundam. Math., Volume 189 (2006), pp. 27-34 | DOI | MR | Zbl
[Ers03] On Isoperimetric Profiles of Finitely Generated Groups, Geom. Dedicata, Volume 100 (2003) no. 1, pp. 157-171 | DOI | MR | Zbl
[EZ20] Isoperimetric Inequalities, Shapes of Følner Sets and Groups with Shalom’s Property , Ann. Inst. Fourier, Volume 70 (2020) no. 4, pp. 1363-1402 | DOI | Numdam | Zbl
[FM98] A rigidity theorem for the solvable Baumslag–Solitar groups, Invent. Math., Volume 131 (1998) no. 2, pp. 419-451 (with an Appendix by Daryl Cooper) | DOI | MR | Zbl
[Fur99] Gromov’s Measure Equivalence and Rigidity of Higher Rank Lattices, Ann. Math., Volume 150 (1999) no. 3, pp. 1059-1081 | DOI | MR | Zbl
[Gab05] Examples of groups that are measure equivalent to the free group, Ergodic Theory Dyn. Syst., Volume 25 (2005) no. 6, pp. 1809-1827 | DOI | MR | Zbl
[Gri85] Degrees of Growth of Finitely Generated Groups and the Theory of Invariant Means, Math. USSR, Izv., Volume 25 (1985) no. 2, pp. 259-300 | DOI | Zbl
[Gri18] Introduction to Analysis on Graphs, University Lecture Series, 71, American Mathematical Society, 2018 | DOI | Zbl
[HMT20] Poincaré Profiles of Groups and Spaces, Rev. Mat. Iberoam., Volume 36 (2020) no. 6, pp. 1835-1886 | DOI | Zbl
[Hum17] A Continuum of Expanders, Fundam. Math., Volume 238 (2017), pp. 143-152 | DOI | MR | Zbl
[IKT09] Subequivalence Relations and Positive-Definite Functions, Groups Geom. Dyn., Volume 3 (2009) no. 4, pp. 579-625 | DOI | MR | Zbl
[Kai85] Examples of noncommutative groups with nontrivial exit-boundary, J. Sov. Math., Volume 28 (1985), pp. 579-591 | DOI | Zbl
[Kid08] The Mapping Class Group from the Viewpoint of Measure Equivalence Theory, Memoirs of the American Mathematical Society, 196, American Mathematical Society, 2008 no. 916 | DOI | MR | Zbl
[Kid10] Measure Equivalence Rigidity of the Mapping Class Group, Ann. Math., Volume 171 (2010) no. 3, pp. 1851-1901 | DOI | MR | Zbl
[KM04] Topics in Orbit Equivalence, Lecture Notes in Mathematics, 1852, Springer, 2004 (QA3 .L28, OCLC: ocm56492332) | DOI | Zbl
[KV83] Random Walks on Discrete Groups: Boundary and Entropy, Ann. Probab., Volume 11 (1983) no. 3, pp. 457-490 | DOI | MR | Zbl
[LM18] On a Measurable Analogue of Small Topological Full Groups, Adv. Math., Volume 332 (2018), pp. 235-286 | DOI | MR | Zbl
[LP21] Poisson Boundaries of Lamplighter Groups: Proof of the Kaimanovich–Vershik Conjecture, J. Eur. Math. Soc., Volume 23 (2021) no. 4, pp. 1133-1160 | DOI | MR | Zbl
[Now07] On Exactness and Isoperimetric Profiles of Discrete Groups, J. Funct. Anal., Volume 243 (2007) no. 1, pp. 323-344 | DOI | MR | Zbl
[OW80] Ergodic Theory of Amenable Group Actions. I. The Rohlin Lemma, Bull. Am. Math. Soc., Volume 2 (1980) no. 1, pp. 161-164 | DOI | MR | Zbl
[OW87] Entropy and Isomorphism Theorems for Actions of Amenable Groups, J. Anal. Math., Volume 48 (1987) no. 1, pp. 1-141 | DOI | MR | Zbl
[Pan89] Métriques de Carnot-Carathéodory et Quasiisométries Des Espaces Symétriques de Rang Un, Ann. Math., Volume 129 (1989) no. 1, pp. 1-60 | DOI | Zbl
[Pit95] Følner Sequences in Polycyclic Groups, Rev. Mat. Iberoam., Volume 11 (1995) no. 3, pp. 675-685 | DOI | MR | Zbl
[Pit00] The Isoperimetric Profile of Homogeneous Riemannian Manifolds, J. Differ. Geom., Volume 54 (2000) no. 2, pp. 255-302 | DOI | MR | Zbl
[Sau02] -Invariants of Groups and Discrete Measured Groupoids, Ph. D. Thesis, Universität Münster, France (2002)
[Sha04] Harmonic Analysis, Cohomology, and the Large-Scale Geometry of Amenable Groups, Acta Math., Volume 192 (2004) no. 2, pp. 119-185 | DOI | MR | Zbl
[Tes08] Large Scale Sobolev Inequalities on Metric Measure Spaces and Applications, Rev. Mat. Iberoam., Volume 24 (2008) no. 3, pp. 825-864 | DOI | MR | Zbl
[Wei01] Monotileable Amenable Groups, Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin’s Memorial, American Mathematical Society, 2001, pp. 257-262 | DOI | Zbl
[Woe05] Lamplighters, Diestel–Leader graphs, random walks, and harmonic functions, Comb. Probab. Comput., Volume 14 (2005) no. 3, pp. 415-433 | DOI | MR | Zbl
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