Volume and homology growth of aspherical manifolds

Sauer, Roman¹

¹ Fakultät für Mathematik, Karlsruhe Institute of Technology, Institut für Algebra und Geometrie (IAG), AG Topologie, Englerstr. 2, D-76131 Karlsruhe, Germany

Geometry & topology, Tome 20 (2016) no. 2, pp. 1035-1059.

Résumé

(1) We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. (2) We show that linear growth of mod $p$ Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is “large”.

Détail

DOI : 10.2140/gt.2016.20.1035

Classification : 53C23, 20F69, 57N65
Keywords: homology growth, aspherical manifolds, residually finite groups

Bibliographie

[1] M Abért, N Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc. (JEMS) 14 (2012) 1657

[2] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, 61, Birkhäuser (1985)

[3] N Bergeron, A Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013) 391

[4] G E Bredon, Introduction to compact transformation groups, 46, Academic Press (1972)

[5] C W Curtis, I Reiner, Methods of representation theory, I, Wiley (1981)

[6] V Emery, Torsion homology of arithmetic lattices and K2 of imaginary fields, Math. Z. 277 (2014) 1155

[7] T Gelander, Homotopy type and volume of locally symmetric manifolds, Duke Math. J. 124 (2004) 459

[8] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982) 5

[9] M Gromov, Large Riemannian manifolds, from: "Curvature and topology of Riemannian manifolds" (editors K Shiohama, T Sakai, T Sunada), Lecture Notes in Math. 1201, Springer (1986) 108

[10] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (2007)

[11] L Guth, Volumes of balls in large Riemannian manifolds, Ann. of Math. 173 (2011) 51

[12] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[13] M Lackenby, Large groups, property (τ) and the homology growth of subgroups, Math. Proc. Cambridge Philos. Soc. 146 (2009) 625

[14] M Lackenby, Detecting large groups, J. Algebra 324 (2010) 2636

[15] W Lück, L2–invariants : theory and applications to geometry and K–theory, 44, Springer (2002)

[16] W Lück, Approximating L2–invariants and homology growth, Geom. Funct. Anal. 23 (2013) 622

[17] D S Ornstein, B Weiss, Ergodic theory of amenable group actions, I : The Rohlin lemma, Bull. Amer. Math. Soc. 2 (1980) 161

[18] R Sauer, Amenable covers, volume and L2–Betti numbers of aspherical manifolds, J. Reine Angew. Math. 636 (2009) 47

[19] C Soulé, Perfect forms and the Vandiver conjecture, J. Reine Angew. Math. 517 (1999) 209

[20] T Tom Dieck, Transformation groups, 8, de Gruyter (1987)

[21] B Weiss, Monotileable amenable groups, from: "Topology, ergodic theory, real algebraic geometry" (editors V Turaev, A Vershik), Amer. Math. Soc. Transl. Ser. 2 202, Amer. Math. Soc. (2001) 257