Torsion Invariants

  • Holger Kammeyer
Part of the Lecture Notes in Mathematics book series (LNM, volume 2247)


We explain how 2-torsion arises as the 2-counterpart to Reidemeister torsion and give an intuitive picture on how to think about chain complexes in this context. We extract some explicit computations of 2-torsion for spaces and groups from the literature and we present how twisting the 2-chain complex with a character leads to the definition of 2-Alexander torsion. As we will see, the degree of the 2-Alexander torsion function recovers the Thurston norm for 3-manifolds. Next we give full details how 2-torsion relates to torsion in homology via regulators. As a consequence, we show how the torsion approximation conjecture actually splits up into three different problems, each of it intriguing and wide open. Specializing to an arithmetic setting, we present the Bergeron Venkatesh conjecture on torsion in twisted homology. We conclude the text with an account on profinite rigidity, (non-)profiniteness of 2-Betti numbers and a proof that the torsion approximation conjecture implies profiniteness of volume for 3-manifolds.


  1. 2.
    M. Abért, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the growth of L 2-invariants for sequences of lattices in Lie groups. Ann. Math. (2) 185(3), 711–790 (2017). MR 3664810MathSciNetzbMATHCrossRefGoogle Scholar
  2. 3.
    I. Agol, Criteria for virtual fibering. J. Topol. 1(2), 269–284 (2008). MR 2399130MathSciNetzbMATHCrossRefGoogle Scholar
  3. 4.
    I. Agol, The virtual Haken conjecture. Doc. Math. 18, 1045–1087 (2013). With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553Google Scholar
  4. 5.
    M. Aka, Profinite completions and Kazhdan’s property (T). Groups Geom. Dyn. 6(2), 221–229 (2012). MR 2914858Google Scholar
  5. 6.
    R.C. Alperin, An elementary account of Selberg’s lemma. Enseign. Math. (2) 33(3–4), 269–273 (1987). MR 925989Google Scholar
  6. 7.
    M. Aschenbrenner, S. Friedl, H. Wilton, 3-Manifold Groups. EMS Series of Lectures in Mathematics (European Mathematical Society, Zürich, 2015). MR 3444187Google Scholar
  7. 11.
    H. Bass, J. Milnor, J.-P Serre, Solution of the congruence subgroup problem for SLn(n ≥ 3) and Sp2n(n ≥ 2). Inst. Hautes Études Sci. Publ. Math. 33, 59–137 (1967). MR 0244257Google Scholar
  8. 15.
    F. Ben Aribi, The L 2-Alexander invariant detects the unknot. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15, 683–708 (2016). MR 3495444Google Scholar
  9. 16.
    N. Bergeron, Torsion homology growth in arithmetic groups, in European Congress of Mathematics (Eur. Math. Soc., Zürich, 2018), pp. 263–287. MR 3887771Google Scholar
  10. 18.
    N. Bergeron, A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12(2), 391–447 (2013). MR 3028790MathSciNetzbMATHCrossRefGoogle Scholar
  11. 21.
    A. Borel, The L 2-cohomology of negatively curved Riemannian symmetric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 95–105 (1985). MR 802471MathSciNetzbMATHCrossRefGoogle Scholar
  12. 22.
    A. Borel, J.-P. Serre, Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1973). Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault. MR 0387495Google Scholar
  13. 23.
    M. Borinsky, K. Vogtmann, The Euler characteristic of Out(F n).
  14. 25.
    M.R. Bridson, A.W. Reid, Profinite rigidity, fibering, and the figure-eight knot (2015). arXiv:1505.07886Google Scholar
  15. 26.
    M.R. Bridson, M.D.E. Conder, A.W. Reid, Determining Fuchsian groups by their finite quotients. Israel J. Math. 214(1), 1–41 (2016). MR 3540604MathSciNetzbMATHCrossRefGoogle Scholar
  16. 28.
    E.J. Brody, The topological classification of the lens spaces. Ann. Math. (2) 71, 163–184 (1960). MR 0116336MathSciNetzbMATHCrossRefGoogle Scholar
  17. 30.
    M. Clay, 2-torsion of free-by-cyclic groups. Q. J. Math. 68(2), 617–634 (2017). MR 3667215Google Scholar
  18. 31.
    D. Crowley, W. Lück, T. Macko, Surgery Theory: Foundations (to appear).
  19. 38.
    J. Dubois, C. Wegner, Weighted L 2-invariants and applications to knot theory. Commun. Contemp. Math. 17(1), 1450010 (2015). MR 3291974zbMATHCrossRefGoogle Scholar
  20. 39.
    J. Dubois, S. Friedl, W. Lück, The L 2-Alexander torsion is symmetric. Algebr. Geom. Topol. 15(6), 3599–3612 (2015). MR 3450772MathSciNetzbMATHCrossRefGoogle Scholar
  21. 40.
    J. Dubois, S. Friedl, W. Lück, The L 2-Alexander torsion of 3-manifolds. J. Topol. 9(3), 889–926 (2016). MR 3551842MathSciNetzbMATHCrossRefGoogle Scholar
  22. 46.
    R.H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds and Related Topics: Proceedings of The University of Georgia Institute, 1961 (Prentice-Hall, Englewood Cliffs, 1962), pp. 120–167. MR 0140099Google Scholar
  23. 47.
    S. Friedl, T. Kim, The Thurston norm, fibered manifolds and twisted Alexander polynomials. Topology 45(6), 929–953 (2006). MR 2263219MathSciNetzbMATHCrossRefGoogle Scholar
  24. 48.
    S. Friedl, T. Kitayama, The virtual fibering theorem for 3-manifolds. Enseign. Math. 60(1–2), 79–107 (2014). MR 3262436MathSciNetzbMATHCrossRefGoogle Scholar
  25. 49.
    S. Friedl, W. Lück, The L 2-torsion function and the Thurston norm of 3-manifolds (2015). arXiv:1510.00264Google Scholar
  26. 50.
    S. Friedl, W. Lück, Universal L 2-torsion, polytopes and applications to 3-manifolds. Proc. Lond. Math. Soc. (3) 114(6), 1114–1151 (2017). MR 3661347MathSciNetzbMATHCrossRefGoogle Scholar
  27. 51.
    S. Friedl, A. Juhász, J. Rasmussen, The decategorification of sutured Floer homology. J. Topol. 4(2), 431–478 (2011). MR 2805998MathSciNetzbMATHCrossRefGoogle Scholar
  28. 52.
    L. Funar, Torus bundles not distinguished by TQFT invariants. Geom. Topol. 17(4), 2289–2344 (2013). With an appendix by Funar and Andrei Rapinchuk. MR 3109869Google Scholar
  29. 58.
    D. Gaboriau, On the top-dimensional -Betti numbers.
  30. 60.
    Ł. Grabowski, Group ring elements with large spectral density. Math. Ann. 363(1–2), 637–656 (2015). MR 3394391MathSciNetzbMATHCrossRefGoogle Scholar
  31. 63.
    M. Gromov, Large Riemannian manifolds, in Curvature and Topology of Riemannian Manifolds (Katata, 1985). Lecture Notes in Mathematics, vol. 1201 (Springer, Berlin, 1986), pp. 108–121. MR 859578CrossRefGoogle Scholar
  32. 69.
    J. Hempel, 3-Manifolds. Annals of Mathematics Studies, No. 86 (Princeton University Press/University of Tokyo Press, Princeton/Tokyo, 1976). MR 0415619Google Scholar
  33. 70.
    J. Hempel, Residual finiteness for 3-manifolds, in Combinatorial Group Theory and Topology (Alta, UT, 1984). Annals of Mathematics Studies, vol. 111 (Princeton University Press, Princeton, 1987), pp. 379–396. MR 895623CrossRefGoogle Scholar
  34. 71.
    J. Hempel, Some 3-manifold groups with the same finite quotients (2014). arXiv:1409.3509Google Scholar
  35. 72.
    G. Herrmann, The L 2-Alexander torsion for Seifert fiber spaces. Arch. Math. (Basel) 109(3), 273–283 (2017). MR 3687871MathSciNetzbMATHCrossRefGoogle Scholar
  36. 73.
    E. Hess, T. Schick, L 2-torsion of hyperbolic manifolds. Manuscripta Math. 97(3), 329–334 (1998). MR 1654784Google Scholar
  37. 84.
    H. Kammeyer, L 2-invariants of nonuniform lattices in semisimple Lie groups. Algebr. Geom. Topol. 14(4), 2475–2509 (2014). MR 3331619MathSciNetzbMATHCrossRefGoogle Scholar
  38. 87.
    H. Kammeyer, The shrinkage type of knots. Bull. Lond. Math. Soc. 49(3), 428–442 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 88.
    H. Kammeyer, A remark on torsion growth in homology and volume of 3-manifolds (2018). arXiv:1802.09244Google Scholar
  40. 89.
    H. Kammeyer, Profinite commensurability of S-arithmetic groups (2018). arXiv:1802.08559Google Scholar
  41. 90.
    H. Kammeyer, R. Sauer, S-arithmetic spinor groups with the same finite quotients and distinct 2-cohomology (2018). arXiv:1804.10604Google Scholar
  42. 91.
    H. Kammeyer, S. Kionke, J. Raimbault, R. Sauer, Profinite invariants of arithmetic groups (2019). arXiv:1901.01227Google Scholar
  43. 95.
    S. Kionke, Lefschetz numbers of symplectic involutions on arithmetic groups. Pacific J. Math. 271(2), 369–414 (2014). MR 3267534MathSciNetzbMATHCrossRefGoogle Scholar
  44. 98.
    M. Kreck, W. Lück, Topological rigidity for non-aspherical manifolds. Pure Appl. Math. Q. 5(3), 873–914 (2009). Special Issue: In honor of Friedrich Hirzebruch. MR 2532709Google Scholar
  45. 104.
    W. Li, W. Zhang, An L 2-Alexander invariant for knots. Commun. Contemp. Math. 8(2), 167–187 (2006). MR 2219611zbMATHCrossRefGoogle Scholar
  46. 108.
    Y. Liu, Degree of L 2-Alexander torsion for 3-manifolds. Invent. Math. 207(3), 981–1030 (2017). MR 3608287Google Scholar
  47. 110.
    J. Lott, Heat kernels on covering spaces and topological invariants. J. Differ. Geom. 35(2), 471–510 (1992). MR 1158345MathSciNetzbMATHCrossRefGoogle Scholar
  48. 111.
    J. Lott, The zero-in-the-spectrum question. Enseign. Math. (2) 42(3–4), 341–376 (1996). MR 1426443Google Scholar
  49. 113.
    J. Lott, W. Lück, L 2-topological invariants of 3-manifolds. Invent. Math. 120(1), 15–60 (1995). MR 1323981Google Scholar
  50. 118.
    W. Lück, Survey on classifying spaces for families of subgroups, in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol. 248 (Birkhäuser, Basel, 2005), pp. 269–322. MR 2195456Google Scholar
  51. 119.
    W. Lück, Twisting L 2-invariants with finite-dimensional representation (2015). arXiv:1510.00057Google Scholar
  52. 120.
    W. Lück, Approximating L 2-invariants by their classical counterparts. EMS Surv. Math. Sci. 3(2), 269–344 (2016). MR 3576534Google Scholar
  53. 122.
    W. Lück, T. Schick, L 2-torsion of hyperbolic manifolds of finite volume. Geom. Funct. Anal. 9(3), 518–567 (1999). MR 1708444Google Scholar
  54. 123.
    W. Lück, R. Sauer, C. Wegner, L 2-torsion, the measure-theoretic determinant conjecture, and uniform measure equivalence. J. Topol. Anal. 2(2), 145–171 (2010). MR 2652905zbMATHCrossRefGoogle Scholar
  55. 125.
    G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17 (Springer, Berlin, 1991). MR 1090825Google Scholar
  56. 126.
    S. Marshall, W. Müller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds. Duke Math. J. 162(5), 863–888 (2013). MR 3047468MathSciNetzbMATHCrossRefGoogle Scholar
  57. 128.
    G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces. Annals of Mathematics Studies, No. 78 (Princeton University Press/University of Tokyo Press, Princeton/Tokyo, 1973). MR 0385004Google Scholar
  58. 129.
    W. Müller, Analytic torsion and R-torsion for unimodular representations. J. Am. Math. Soc. 6(3), 721–753 (1993). MR 1189689Google Scholar
  59. 130.
    W. Müller, J. Pfaff, Analytic torsion of complete hyperbolic manifolds of finite volume. J. Funct. Anal. 263(9), 2615–2675 (2012). MR 2967302MathSciNetzbMATHCrossRefGoogle Scholar
  60. 133.
    W. Müller, J. Pfaff, On the growth of torsion in the cohomology of arithmetic groups. Math. Ann. 359(1–2), 537–555 (2014). MR 3201905MathSciNetzbMATHCrossRefGoogle Scholar
  61. 134.
    W. Müller, J. Pfaff, The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume. J. Funct. Anal. 267(8), 2731–2786 (2014). MR 3255473MathSciNetzbMATHCrossRefGoogle Scholar
  62. 136.
    N. Nikolov, Algebraic properties of profinite groups (2011). arXiv:1108.5130Google Scholar
  63. 137.
    N. Nikolov, D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds. Ann. Math. (2) 165(1), 171–238 (2007). MR 2276769MathSciNetzbMATHCrossRefGoogle Scholar
  64. 139.
    M. Olbrich, L 2-invariants of locally symmetric spaces. Doc. Math. 7, 219–237 (2002). MR 1938121Google Scholar
  65. 146.
    V. Platonov, A. Rapinchuk, Algebraic Groups and Number Theory. Pure and Applied Mathematics, vol. 139 (Academic, Boston, 1994). Translated from the 1991 Russian original by Rachel Rowen. MR 1278263zbMATHCrossRefGoogle Scholar
  66. 147.
    G. Prasad, A.S. Rapinchuk, Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre (2008). arXiv:0809.1622Google Scholar
  67. 148.
    P. Przytycki, D.T. Wise, Mixed 3-manifolds are virtually special. J. Am. Math. Soc. 31(2), 319–347 (2018). MR 3758147Google Scholar
  68. 149.
    A.A. Ranicki, Notes on Reidemeister Torsion. Department of Mathematics and Statistics University of Edinburgh.
  69. 151.
    A.W. Reid, Profinite properties of discrete groups, in Groups St. Andrews 2013. London Mathematics Society. Lecture Note Series, vol. 422 (Cambridge University Press, Cambridge, 2015), pp. 73–104. MR 3445488Google Scholar
  70. 152.
    L. Ribes, P. Zalesskii, Profinite Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 40 (Springer, Berlin, 2000). MR 1775104Google Scholar
  71. 160.
    P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. Math. (2) 182(3), 945–1066 (2015). MR 3418533Google Scholar
  72. 162.
    M.H. Şengün, On the integral cohomology of Bianchi groups. Exp. Math. 20(4), 487–505 (2011). MR 2859903MathSciNetzbMATHCrossRefGoogle Scholar
  73. 163.
    M.H. Şengün, On the torsion homology of non-arithmetic hyperbolic tetrahedral groups. Int. J. Number Theory 8(2), 311–320 (2012). MR 2890481MathSciNetzbMATHCrossRefGoogle Scholar
  74. 165.
    W. Thurston, The Geometry and Topology of 3-Manifolds. Lecture Notes (1980).
  75. 166.
    W. Thurston, A norm for the homology of 3-manifolds. Mem. Am. Math. Soc. 59(339), i–vi and 99–130 (1986). MR 823443Google Scholar
  76. 168.
    K. Vogtmann, Automorphisms of free groups and outer space, in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 1–31. MR 1950871Google Scholar
  77. 170.
    H.C. Wang, Topics on totally discontinuous groups, in Symmetric Spaces (Short Courses, Washington University, St. Louis, 1969–1970). Pure and Applied Mathematics, vol. 8 (Dekker, New York, 1972), pp. 459–487. MR 0414787Google Scholar
  78. 172.
    G. Wilkes, Profinite rigidity for Seifert fibre spaces. Geom. Dedicata 188, 141–163 (2017). MR 3639628MathSciNetzbMATHCrossRefGoogle Scholar
  79. 174.
    D.T. Wise, The structure of groups with a quasiconvex hierarchy (2012).
  80. 175.
    D. Witte Morris, Introduction to Arithmetic Groups (Deductive Press, Public Domain, 2015). MR 3307755Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Holger Kammeyer
    • 1
  1. 1.Institute for Algebra and GeometryKarlsruhe Institute of TechnologyKarlsruheGermany

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