Growth of homology torsion in finite coverings and hyperbolic volume
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3173
Classification: 57M27, 57M25
Keywords: Homology torsion, covering, Fuglede-Kadison determinant, hyperbolic volume
Mot clés : torsion homologique, revêtements, déterminant de Fuglede-Kadison, volume hyperbolique
Lê, Thang T. Q. 1

1 School of Mathematics 686 Cherry Street, Georgia Tech Atlanta, GA 30332 (USA)
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Lê, Thang T. Q. 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. http://www.numdam.org/articles/10.5802/aif.3173/

[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, Lectures in Mathematics, Birkhäuser, 2008, x+334 pages | MR | Zbl

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

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

[5] Bergeron, Nicolas; Venkatesh, Akshay The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu, Volume 12 (2013) no. 2, pp. 391-447 | 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, x+237 pages | DOI | MR | Zbl

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

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

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

[10] Farber, Michael Geometry of growth: approximation theorems for L 2 invariants, Math. Ann., Volume 311 (1998) no. 2, pp. 335-375 | DOI | MR | Zbl

[11] Gordon, C. McA. Knots whose branched cyclic coverings have periodic homology, Trans. Am. Math. Soc., Volume 168 (1972), pp. 357-370 | DOI | MR | Zbl

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

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

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

[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 | Zbl

[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., Volume 89 (2014) no. 3, pp. 719-757 | DOI | MR | Zbl

[18] 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

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

[20] Lück, Wolfgang L 2 -invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages | DOI | MR | Zbl

[21] Lück, Wolfgang Approximating L 2 -invariants and homology growth, Geom. Funct. Anal., Volume 23 (2013) no. 2, pp. 622-663 | DOI | MR | Zbl

[22] Lück, Wolfgang; Schick, Thomas L 2 -torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 518-567 | DOI | MR | Zbl

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

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

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

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

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

[28] Turaev, Vladimir Introduction to combinatorial torsions, Lectures in Mathematics, Birkhäuser, 2001, viii+123 pages | DOI | MR | Zbl

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