# ANNALES DE L'INSTITUT FOURIER

Growth of homology torsion in finite coverings and hyperbolic volume
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, p. 611-645
We give an upper bound for the growth of homology torsions of finite coverings of irreducible oriented 3-manifolds in terms of the hyperbolic volume.
Nous donnons une limite supérieure pour la croissance des torsions homologiques de revêtements finis de 3-variétés orientées irréductibles en termes du volume hyperbolique.
Revised : 2017-05-08
Accepted : 2017-07-13
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3173
Classification:  57M27,  57M25
Keywords: Homology torsion, covering, Fuglede-Kadison determinant, hyperbolic volume
@article{AIF_2018__68_2_611_0,
author = {L\^e, Thang T. Q.},
title = {Growth of homology torsion in finite coverings and hyperbolic volume},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {68},
number = {2},
year = {2018},
pages = {611-645},
doi = {10.5802/aif.3173},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_2018__68_2_611_0}
}

Growth of homology torsion in finite coverings and hyperbolic volume. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 611-645. doi : 10.5802/aif.3173. https://aif.centre-mersenne.org/item/AIF_2018__68_2_611_0/

[1] Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo On the growth of ${L}^{2}$-invariants for sequences of lattices in Lie groups (2012) (https://arxiv.org/abs/1210.2961 )

[2] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, Lectures in Mathematics (2008), x+334 pages | MR 2401600 | Zbl 1145.35001

[3] Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry 3-manifold groups, European Mathematical Society, EMS Series of Lectures in Mathematics (2015), xiv+215 pages | Article | MR 3444187 | Zbl 1326.57001

[4] Bergeron, Nicolas; Şengün, Mehmet Haluk; Venkatesh, Akshay Torsion homology growth and cycle complexity of arithmetic manifolds, Duke Math. J., Tome 165 (2016) no. 9, pp. 1629-1693 | Article | MR 3513571 | Zbl 1351.11031

[5] Bergeron, Nicolas; Venkatesh, Akshay The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu, Tome 12 (2013) no. 2, pp. 391-447 | Article | MR 3028790 | Zbl 1266.22013

[6] Bessières, Laurent; Besson, Gérard; Maillot, Sylvain; Boileau, Michel; Porti, Joan Geometrisation of 3-manifolds, European Mathematical Society, EMS Tracts in Mathematics, Tome 13 (2010), x+237 pages | Article | MR 2683385 | Zbl 1244.57003

[7] Boileau, Michel Thick/thin decomposition of three-manifolds and the geometrisation conjecture, Ricci flow and geometric applications, Springer (Lecture Notes in Math.) Tome 2166 (2016), pp. 21-70 | MR 3615962

[8] Bowen, Lewis Measure conjugacy invariants for actions of countable sofic groups, J. Am. Math. Soc., Tome 23 (2010) no. 1, pp. 217-245 | Article | MR 2552252 | Zbl 1201.37005

[9] Brock, Jeffrey F.; Dunfield, Nathan M. Injectivity radii of hyperbolic integer homology 3-spheres, Geom. Topol., Tome 19 (2015) no. 1, pp. 497-523 | Article | MR 3318758 | Zbl 1312.57022

[10] Farber, Michael Geometry of growth: approximation theorems for ${L}^{2}$ invariants, Math. Ann., Tome 311 (1998) no. 2, pp. 335-375 | Article | MR 1625742 | Zbl 0911.53026

[11] Gordon, C. Mca. Knots whose branched cyclic coverings have periodic homology, Trans. Am. Math. Soc., Tome 168 (1972), pp. 357-370 | Article | MR 0295327 | Zbl 0238.55001

[12] Gromov, Mikhael Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991), Cambridge University Press (London Mathematical Society Lecture Note Series) Tome 182 (1993), pp. 1-295 | MR 1253544 | Zbl 0841.20039

[13] Hempel, John $3$-Manifolds, Princeton University Press; University of Tokyo Press, Annals of Mathematics Studies, Tome 86 (1976), xii+195 pages | MR 0415619 | Zbl 0345.57001

[14] Hempel, John Residual finiteness for $3$-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984), Princeton University Press (Annals of Mathematics Studies) Tome 111 (1987), pp. 379-396 | MR 895623 | Zbl 0772.57002

[15] Kajdan, D. A. On arithmetic varieties, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York (1975), pp. 151-217 | MR 0486316 | Zbl 0308.14007

[16] Lê, Thang T. Q. Hyperbolic volume, Mahler measure, and homology growth (talk at Columbia University (2009), slide available at http://www.math.columbia.edu/~volconf09/notes/leconf.pdf)

[17] Lê, Thang T. Q. Homology torsion growth and Mahler measure, Comment. Math. Helv., Tome 89 (2014) no. 3, pp. 719-757 | Article | MR 3260847 | Zbl 1302.57005

[18] Lück, Wolfgang Approximating ${L}^{2}$-invariants by their finite-dimensional analogues, Geom. Funct. Anal., Tome 4 (1994) no. 4, pp. 455-481 | Article | MR 1280122 | Zbl 0853.57021

[19] Lück, Wolfgang ${L}^{2}$-torsion and $3$-manifolds, Low-dimensional topology (Knoxville, TN, 1992), International Press (Conference Proceedings and Lecture Notes in Geometry and Topology) Tome 3 (1994), pp. 75-107 | MR 1316175 | Zbl 0870.57029

[20] Lück, Wolfgang ${L}^{2}$-invariants: theory and applications to geometry and $K$-theory, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 44 (2002), xvi+595 pages | Article | MR 1926649 | Zbl 1009.55001

[21] Lück, Wolfgang Approximating ${L}^{2}$-invariants and homology growth, Geom. Funct. Anal., Tome 23 (2013) no. 2, pp. 622-663 | Article | MR 3053758 | Zbl 1273.22009

[22] Lück, Wolfgang; Schick, Thomas ${L}^{2}$-torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal., Tome 9 (1999) no. 3, pp. 518-567 | Article | MR 1708444 | Zbl 0947.58024

[23] Marshall, Simon; Müller, Werner On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J., Tome 162 (2013) no. 5, pp. 863-888 | Article | MR 3047468 | Zbl 1316.11042

[24] Menal-Ferrer, Pere; Porti, Joan Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol., Tome 7 (2014) no. 1, pp. 69-119 | Article | MR 3180614 | Zbl 1302.57044

[25] Müller, Werner The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, Springer (Progress in Mathematics) Tome 297 (2012), pp. 317-352 | Article | MR 3220447 | Zbl 1264.58026

[26] Rourke, Colin P.; Sanderson, Brian J. Introduction to piecewise-linear topology, Springer Tome 69 (1972), viii+123 pages (Ergebnisse der Mathematik und ihrer Grenzgebiete) | MR 0350744 | Zbl 0254.57010

[27] Sauer, Roman Volume and homology growth of aspherical manifolds, Geom. Topol., Tome 20 (2016) no. 2, pp. 1035-1059 | Article | MR 3493098 | Zbl 1338.53067

[28] Turaev, Vladimir Introduction to combinatorial torsions, Birkhäuser, Lectures in Mathematics (2001), viii+123 pages | Article | MR 1809561 | Zbl 0970.57001