On the top-dimensional ${\ell }^2$-Betti numbers

Gaboriau, Damien; Noûs, Camille

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 5, pp. 1121-1137.

Le but de cette note est d’introduire une astuce qui relie l’annulation (ou la non-annulation) du nombre de Betti ${\ell }^2$ en dimension maximale des actions d’un groupe avec l’annulation pour ses sous-actions. On fournit trois différents types d’applications : on montre que les nombres de Betti ${\ell }^2$ de $\mathrm{Aut}\left({\mathbf{F}}_n\right)$ et $\mathrm{Out}\left({\mathbf{F}}_n\right)$ (et de leurs sous-groupes de Torelli) ne s’annulent pas en degré égal à leur dimension cohomologique virtuelle ; on prouve qu’un sous-groupe quelconque du groupe fondamental d’une variété compacte de dimension $3$ a ses nombres de Betti ${\ell }^2$ nuls en degré $3$ et $2$ et enfin, on parvient à déterminer la dimension ergodique de certains produits directs de la forme $H\times A$ où $A$ est moyennable infini.

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional ${\ell }^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the ${\ell }^2$-Betti numbers of $\mathrm{Aut}\left({\mathbf{F}}_n\right)$ and $\mathrm{Out}\left({\mathbf{F}}_n\right)$ (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the $3$-manifold groups have vanishing ${\ell }^2$-Betti numbers in degree $3$ and $2$ and we figure out the ergodic dimension of certain direct products of the form $H\times A$ where $A$ is infinite amenable.

Classification : 37A20, 19K56, 20F28, 20E15, 57MXX
Mots clés : $\ell ^2$-Betti numbers, measured group theory, cohomological dimension, ergodic dimension, $\mathrm{Out}(\mathbf{F}_n), \mathrm{Aut}(\mathbf{F}_n)$, $3$-dimensional manifolds.

[1] Abért, M.; Gaboriau, Damien Higher dimensional cost and profinite actions, 2022 (in preparation)

[2] Atiyah, Michael F. Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) (Astérisque), Volume 32-33, Société Mathématique de France, 1974, pp. 43-72 | Zbl

[3] Bartholdi, Laurent The rational homology of the outer automorphism group of $F_7$, New York J. Math., Volume 22 (2016), pp. 191-197 | MR | Zbl

[4] Bartholdi, Laurent; Gaboriau, Damien Around the homology of $\mathrm{Out}\left({𝔽}_n\right)$, 2022 (in preparation)

[5] Baumslag, Gilbert Automorphism groups of residually finite groups, J. Lond. Math. Soc., Volume 38 (1963), pp. 117-118 | DOI | MR | Zbl

[6] Bessières, Laurent; Besson, Gérard; Maillot, Sylvain; Boileau, Michel; Porti, Joan Geometrisation of 3-manifolds, EMS Tracts in Mathematics, 13, European Mathematical Society, 2010 | DOI

[7] Bestvina, Mladen; Bux, Kai-Uwe; Margalit, Dan Dimension of the Torelli group for $\mathrm{Out}\left(F_n\right)$, Invent. Math., Volume 170 (2007) no. 1, pp. 1-32 | DOI | MR | Zbl

[8] Bestvina, Mladen; Kapovich, Michael; Kleiner, Bruce Van Kampen’s embedding obstruction for discrete groups, Invent. Math., Volume 150 (2002) no. 2, pp. 219-235 | DOI | MR | Zbl

[9] Borel, Armand The $L_2$-cohomology of negatively curved Riemannian symmetric spaces, Ann. Acad. Sci. Fenn., Math., Volume 10 (1985), pp. 95-105 | DOI | MR | Zbl

[10] Borinsky, Michael; Vogtmann, Karen The Euler characteristic of $\mathrm{Out}\left(F_n\right)$, Comment. Math. Helv., Volume 95 (2020) no. 4, pp. 703-748 | DOI | MR | Zbl

[11] Bridson, Martin R.; Vogtmann, Karen Automorphism groups of free groups, surface groups and free abelian groups, Problems on mapping class groups and related topics (Proceedings of Symposia in Pure Mathematics), Volume 74, American Mathematical Society, 2006, pp. 301-316 | DOI | MR | Zbl

[12] Brown, Kenneth S. Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1982 | DOI

[13] Cheeger, Jeff; Gromov, Mikhael $L_2$-cohomology and group cohomology, Topology, Volume 25 (1986) no. 2, pp. 189-215 | DOI | MR | Zbl

[14] Conley, Clinton T.; Gaboriau, Damien; Marks, Andrew S.; Tucker-Drob, Robin D. One-ended spanning subforests and treeability of groups (2021) (https://arxiv.org/abs/2104.07431)

[15] Connes, Alain Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) (Lecture Notes in Mathematics), Volume 725, Springer, 1978, pp. 19-143 | DOI | Zbl

[16] Culler, Marc; Vogtmann, Karen Moduli of graphs and automorphisms of free groups, Invent. Math., Volume 84 (1986) no. 1, pp. 91-119 | DOI | MR | Zbl

[17] Dixmier, Jacques Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Cahiers Scientifiques, XXV, Gauthier-Villars, 1969 | Numdam

[18] Dodziuk, Jozef $L^2$ harmonic forms on rotationally symmetric Riemannian manifolds, Proc. Am. Math. Soc., Volume 77 (1979), pp. 395-400 | MR | Zbl

[19] Eckmann, Beno Introduction to ${\ell }_2$-methods in topology: reduced ${\ell }_2$-homology, harmonic chains, ${\ell }_2$-Betti numbers, Isr. J. Math., Volume 117 (2000), pp. 183-219 | DOI | Zbl

[20] Eilenberg, Samuel; Ganea, Tudor On the Lusternik–Schnirelmann category of abstract groups, Ann. Math., Volume 65 (1957), pp. 517-518 | DOI | MR | Zbl

[21] Gaboriau, Damien Invariants ${\ell }_2$ de relations d’équivalence et de groupes, Publ. Math., Volume 95 (2002), pp. 93-150 | DOI | Zbl

[22] Gaboriau, Damien On the ergodic dimension (2022) (in preparation)

[23] Gaboriau, Damien; Lyons, Russell A measurable-group-theoretic solution to von Neumann’s problem, Invent. Math., Volume 177 (2009) no. 3, pp. 533-540 | DOI | MR | Zbl

[24] Grossman, Edna K. On the residual finiteness of certain mapping class groups, J. Lond. Math. Soc., Volume 9 (1974), pp. 160-164 | DOI | MR | Zbl

[25] Hatcher, Allen E. Homological stability for automorphism groups of free groups, Comment. Math. Helv., Volume 70 (1995) no. 1, pp. 39-62 | DOI | MR | Zbl

[26] Kammeyer, Holger Introduction to ${\ell }^2$-invariants, Lecture Notes in Mathematics, 2247, Springer, 2019 | DOI | MR

[27] Kida, Yoshikata The mapping class group from the viewpoint of measure equivalence theory, Memoirs of the American Mathematical Society, 916, American Mathematical Society, 2008, viii+190 pages | MR

[28] Kleiner, Bruce; Lott, John Notes on Perelman’s papers, Geom. Topol., Volume 12 (2008) no. 5, pp. 2587-2855 | DOI | MR | Zbl

[29] Kneser, Hellmuth Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresber. Dtsch. Math.-Ver., Volume 38 (1929), pp. 248-259

[30] Lott, John; Lück, Wolfgang $L_2$-topological invariants of $3$-manifolds, Invent. Math., Volume 120 (1995) no. 1, pp. 15-60 | DOI | MR | Zbl

[31] Lück, Wolfgang Approximating $L^2$-invariants by their finite-dimensional analogues, Geom. Funct. Anal., Volume 4 (1994) no. 4, pp. 455-481 | DOI | MR | Zbl

[32] Lück, Wolfgang Dimension theory of arbitrary modules over finite von Neumann algebras and $L_2$-Betti numbers. II. Applications to Grothendieck groups, $L_2$-Euler characteristics and Burnside groups, J. Reine Angew. Math. (1998), pp. 213-236 | DOI | MR | Zbl

[33] Lück, Wolfgang $L_2$-invariants: theory and applications to geometry and $K$-theory, Theory and applications, 44, Springer, 2002

[34] Magnus, Wilhelm Über $n$-dimensionale Gittertransformationen, Acta Math., Volume 64 (1935), pp. 353-367 | DOI | Zbl

[35] Milnor, John W. A unique decomposition theorem for $3$-manifolds, Am. J. Math., Volume 84 (1962), pp. 1-7 | DOI | MR | Zbl

[36] Nielsen, Jakob Die Isomorphismengruppe der freien Gruppen, Math. Ann., Volume 91 (1924) no. 3-4, pp. 169-209 | DOI | MR | Zbl

[37] Ohashi, Ryo The rational homology group of $\mathrm{Out}\left(F_n\right)$ for $n\le 6$, Exp. Math., Volume 17 (2008) no. 2, pp. 167-179 | DOI | MR | Zbl

[38] Ornstein, Donald S.; Weiss, Benjamin 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

[39] Perelman, Grisha The entropy formula for the Ricci flow and its geometric applications (2002) (https://arxiv.org/abs/math/0211159)

[40] Perelman, Grisha Ricci flow with surgery on three-manifolds (2003) (https://arxiv.org/abs/math/0303109)

[41] Sauer, Roman; Thom, Andreas A spectral sequence to compute $L^2$-Betti numbers of groups and groupoids, J. Lond. Math. Soc., Volume 81 (2010) no. 3, pp. 747-773 | DOI | MR | Zbl

[42] Serre, Jean-Pierre Arbres, amalgames, ${\mathrm{SL}}_2$, Astérisque, 46, Société Mathématique de France, 1977

[43] Vogtmann, Karen The cohomology of automorphism groups of free groups, International Congress of Mathematicians. Vol. II, European Mathematical Society, 2006, pp. 1101-1117 | Zbl