Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-27T15:41:12.323Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic growth of torsion homology for arithmetic groups

Published online by Cambridge University Press:  07 June 2012

Nicolas Bergeron  and
Akshay Venkatesh
Nicolas Bergeron
Affiliation:
Institut de Mathématiques de Jussieu, Unité Mixte de Recherche 7586 du CNRS, Université Pierre et Marie Curie, 4, place Jussieu 75252 Paris Cedex 05, France ([email protected]) URL: http://people.math.jussieu.fr/~bergeron
Akshay Venkatesh
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94304, USA ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

When does the amount of torsion in the homology of an arithmetic group grow exponentially with the covolume? We give many examples where this is the case, and conjecture precise conditions.

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 2 , April 2013 , pp. 391 - 447
Copyright
©Cambridge University Press 2012

References

Agol, Ian, Criteria for virtual fibering, J. Topol. 1 (2) (2008), 269–284.Google Scholar
Ash, Avner and Sinnott, Warren, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod $p$ cohomology of $\mathrm{GL} (n, \mathbb{Z} )$, Duke Math. J. 105 (2000).CrossRefGoogle Scholar
Doud, Darrin, Ash, Avner and Pollack, David, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), 521–579.Google Scholar
Barbasch, Dan and Moscovici, Henri, ${L}^{2} $-index and the Selberg trace formula, J. Funct. Anal. 53 (2) (1983), 151–201.Google Scholar
Bergeron, Nicolas, Haglund, Frédéric and Wise, Daniel T., Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. (2) 83 (2) (2011), 431–448.Google Scholar
Bergeron, Nicolas and Wise, Daniel T., 2012 A boundary criterion for cubulation. Amer. J. Math. 134(3), in press.Google Scholar
Bhargava, Manjul, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. IMRN  (17) (2007), 20 Art. ID rnm052.Google Scholar
Bismut, Jean-Michel and Zhang, Weiping, An extension of a theorem by Cheeger and Müller, Astérisque  (205) (1992), 235. With an appendix by François Laudenbach.Google Scholar
Borel, A., Labesse, J.-P. and Schwermer, J., On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields, Compos. Math. 102 (1) (1996), 1–40.Google Scholar
Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, Mathematical Surveys and Monographs, Volume 67 (American Mathematical Society, Providence, RI, 2000).Google Scholar
Bourbaki, Nicolas, Lie groups and Lie algebras. Chapters 1–3, in Elements of mathematics (Berlin). (Springer-Verlag, Berlin, 1998); Translated from the French, Reprint of the 1989 English translation.Google Scholar
Boyer, Pascal, Pour $l\not = 2$, la ${ \mathbb{Z} }_{l} $-cohomologie du modèle de Deligne-Carayol et de quelques variétés simples de Shimura de Kottwitz est sans torsion, preprint.Google Scholar
Buzzard, Kevin, Diamond, Fred and Jarvis, Frazer, On Serre’s conjecture for mod $\ell $ Galois representations over totally real fields, Duke Math. J. 155 (1) (2010), 105–161.Google Scholar
Buzzard, Kevin and Gee, Toby, 2011 The conjectural connections between automorphic representations and galois representations. In Proceedings of the LMS Durham Symposium, in press.Google Scholar
Calegari, Frank and Dunfield, Nathan M., Automorphic forms and rational homology 3-spheres, Geom. Topol. 10 (2006), 295–329 (electronic).Google Scholar
Calegari, Frank and Emerton, Matthew, Completed cohomology, a survey. Non-abelian Fundamental Groups and Iwasawa Theory, CUP, in press.Google Scholar
Calegari, Frank and Emerton, Matthew, Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms, Ann. of Math. (2) 170 (3) (2009), 1437–1446.CrossRefGoogle Scholar
Calegari, Frank and Emerton, Matthew, Mod-$p$ cohomology growth in $p$-adic analytic towers of 3-manifolds, Groups Geom. Dyn. 5 (2) (2011), 355–366.Google Scholar
Calegari, Frank and Mazur, Barry, Nearly ordinary Galois deformations over arbitrary number fields, J. Inst. Math. Jussieu 8 (1) (2009), 99–177.Google Scholar
Calegari, Frank and Venkatesh, Akshay, Toward a torsion Jacquet–Langlands correspondence, in preparation.Google Scholar
Cheeger, Jeff, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (2) (1979), 259–322.Google Scholar
Cheng, Siu Yuen, Li, Peter and Yau, Shing Tung, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (5) (1981), 1021–1063.Google Scholar
Clair, Bryan and Whyte, Kevin, Growth of Betti numbers, Topology 42 (5) (2003), 1125–1142.CrossRefGoogle Scholar
Cohen, H. and Lenstra, H. W. Jr., Heuristics on class groups of number fields, in Number theory. (Noordwijkerhout, 1983).Google Scholar
Dimitrov, Mladen, On Ihara’s lemma for Hilbert modular varieties, Compos. Math. 145 (5) (2009), 1114–1146.Google Scholar
Dunfield, Nathan M. and Thurston, William P., Finite covers of random 3-manifolds, Invent. Math. 166 (3) (2006), 457–521.CrossRefGoogle Scholar
Elstrodt, J., Grunewald, F. and Mennicke, J., $\mathrm{PSL} (2)$ over imaginary quadratic integers, in Arithmetic Conference (Metz, 1981), Astérisque, Volume 94. pp. 43–60 (Soc. Math., France, Paris, 1982).Google Scholar
Emerton, Matthew, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math. 164 (1) (2006), 1–84.CrossRefGoogle Scholar
Everest, Graham and Ward, Thomas, Heights of polynomials and entropy in algebraic dynamics, in Universitext. (Springer-Verlag London Ltd., London, 1999).Google Scholar
Figueiredo, L. M., Serre’s conjecture for imaginary quadratic fields, Compos. Math. 118 (1) (1999), 103–122.CrossRefGoogle Scholar
Gel’fond, A. O., Transcendental and algebraic numbers (Dover Publications Inc., New York, 1960), Translated from the first Russian edition by Leo F. Boron.Google Scholar
Gross, Benedict, Odd galois representations, preprint (http://www.math.harvard.edu/~gross/preprints/Galois_Rep.pdf).Google Scholar
Gross, Benedict H., On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., Volume 254. pp. 223–237 (Cambridge Univ. Press, Cambridge, 1998).Google Scholar
Grunewald, F., Jaikin-Zapirain, A. and Zalesskii, P. A., Cohomological goodness and the profinite completion of Bianchi groups, Duke Math. J. 144 (1) (2008), 53–72.Google Scholar
Harish-Chandra, , Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104–204.Google Scholar
Harish-Chandra, , Harmonic analysis on real reductive groups. III. The Maass–Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1) (1976), 117–201.Google Scholar
Hecht, Henryk and Schmid, Wilfried, A proof of Blattner’s conjecture, Invent. Math. 31 (2) (1975), 129–154.Google Scholar
Herzig, Florian, The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations, Duke Math. J. 149 (1) (2009), 37–116.Google Scholar
Knapp, Anthony W., Representation theory of semisimple groups, in Princeton landmarks in mathematics (Princeton University Press, Princeton, NJ, 2001), An overview based on examples, Reprint of the 1986 original.Google Scholar
Knieper, G., On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. 7 (4) (1997), 755–782.CrossRefGoogle Scholar
Kochloukova, Dessislava H. and Zalesskii, Pavel A., Profinite and pro-$p$ completions of Poincaré duality groups of dimension 3, Trans. Amer. Math. Soc. 360 (4) (2008), 1927–1949.Google Scholar
Kowalski, Emmanuel, The large sieve, property (T) and the homology of dunfield-thurston random 3-manifolds (unpublished note available at http://www.ufr-mi.u-bordeaux.fr/~kowalski/notes-unpublished.html).Google Scholar
Le, Thang, Homology torsion growth and mahler measure (arXiv:1010.4199).Google Scholar
Li, Weiping and Zhang, Weiping, An ${L}^{2} $-Alexander invariant for knots, Commun. Contemp. Math. 8 (2) (2006), 167–187.Google Scholar
Lind, D. A., Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynam. Syst. 2 (1) (1982), 49–68.Google Scholar
Lott, John, Heat kernels on covering spaces and topological invariants, J. Differential Geom. 35 (2) (1992), 471–510.Google Scholar
Lubotzky, A. and Weiss, B., Groups and expanders, in Expanding graphs (Princeton, NJ, 1992), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Volume 10, pp. 95–109. (Amer. Math. Soc., Providence, RI, 1993).Google Scholar
Lück, W., Approximating ${L}^{2} $-invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (4) (1994), 455–481.Google Scholar
Lück, Wolfgang, ${L}^{2} $-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Volume 44 (Springer-Verlag, Berlin, 2002).Google Scholar
Macdonald, I. G., The volume of a compact Lie group, Invent. Math. 56 (2) (1980), 93–95.Google Scholar
Marshall, Simon and Müller, Werner, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds (arXiv:1103.2262).Google Scholar
Milnor, John, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147.Google Scholar
Müller, Werner, The asymptotics of the ray-singer analytic torsion of hyperbolic 3-manifolds (arXiv:1003:5168).Google Scholar
Müller, Werner, Analytic torsion and $R$-torsion of Riemannian manifolds, Adv. Math. 28 (3) (1978), 233–305.Google Scholar
Müller, Werner, Analytic torsion and $R$-torsion for unimodular representations, J. Amer. Math. Soc. 6 (3) (1993), 721–753.Google Scholar
Neukirch, Jürgen, Schmidt, Alexander and Wingberg, Kay, Cohomology of number fields, second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 323 (Springer-Verlag, Berlin, 2008).Google Scholar
Olbrich, Martin, ${L}^{2} $-invariants of locally symmetric spaces, Doc. Math. 7 (2002), 219–237 (electronic).Google Scholar
Parthasarathy, R., Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. Sect. A Math. Sci. 89 (1) (1980), 1–24.Google Scholar
Raimbault, Jean, Exponential growth of torsion in abelian coverings. Algebraic and Geometric Topology, in press (arXiv:1012.3666).Google Scholar
Ray, D. B. and Singer, I. M., $R$-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971), 145–210.Google Scholar
Sarnak, Peter, Arithmetic quantum chaos. 1992 Schur lectures.Google Scholar
Seeley, R. T., Complex powers of an elliptic operator, in Singular integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 288–307. (Amer. Math. Soc., Providence, RI, 1967).Google Scholar
Şengün, Mehmet Haluk, On the integral cohomology of Bianchi groups, Exp. Math. 20 (4) (2011), 487–505.Google Scholar
Serre, Jean-Pierre, Cohomologie galoisienne, fifth edition, Lecture Notes in Mathematics, Volume 5 (Springer-Verlag, Berlin, 1994).Google Scholar
Silver, Daniel S. and Williams, Susan G., Mahler measure, links and homology growth, Topology 41 (5) (2002), 979–991.CrossRefGoogle Scholar
Silver, Daniel S. and Williams, Susan G., Torsion numbers of augmented groups with applications to knots and links, Enseign. Math. (2) 48 (3–4) (2002), 317–343.Google Scholar
Taylor, Richard, On Congruences between modular forms. PhD thesis (available on http://www.math.harvard.edu/~rtaylor/).Google Scholar
Vogan, David A. Jr. and Zuckerman, Gregg J., Unitary representations with nonzero cohomology, Compos. Math. 53 (1) (1984), 51–90.Google Scholar
Wallach, Nolan R., Real reductive groups. I, Pure and Applied Mathematics, Volume 132 (Academic Press Inc., Boston, MA, 1988).Google Scholar