The torsion in symmetric powers on congruence subgroups of Bianchi groups

Pfaff, Jonathan; Raimbault, Jean

doi:10.1090/tran/7875

The torsion in symmetric powers on congruence subgroups of Bianchi groups
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Trans. Amer. Math. Soc. 373 (2020), 109-148 Request permission

Abstract:

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated with the $m$th symmetric power of the standard representation of $\operatorname {SL}_2(\mathbb {C})$ grows exponentially in $m^2$. We give upper and lower bounds for the growth rate. Our result extends a result of W. Müller and S. Marshall, who proved the corresponding statement for closed arithmetic $3$-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for combinatorial Reidemeister torsions on higher-dimensional hyperbolic manifolds.

References

Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, and Iddo Samet, On the growth of $L^2$-invariants for sequences of lattices in Lie groups, Ann. of Math. (2) 185 (2017), no. 3, 711–790. MR 3664810, DOI 10.4007/annals.2017.185.3.1
Tobias Berger, Denominators of Eisenstein cohomology classes for $\textrm {GL}_2$ over imaginary quadratic fields, Manuscripta Math. 125 (2008), no. 4, 427–470. MR 2392880, DOI 10.1007/s00229-007-0139-6
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Nicolas Bergeron and Akshay Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), no. 2, 391–447. MR 3028790, DOI 10.1017/S1474748012000667
Leslie Cohn, Analytic theory of the Harish-Chandra $C$-function, Lecture Notes in Mathematics, Vol. 429, Springer-Verlag, Berlin-New York, 1974. MR 0422509, DOI 10.1007/BFb0064335
F. Calegari and A. Venkatesh, A torsion Jacquet–Langlands correspondence, arXiv:1212.3847 (2012); Astérisque 409 (to appear).
R. M. Damerell, $L$-functions of elliptic curves with complex multiplication. I, Acta Arith. 17 (1970), 287–301. MR 285540, DOI 10.4064/aa-17-3-287-301
R. M. Damerell, $L$-functions of elliptic curves with complex multiplication. II, Acta Arith. 19 (1971), 311–317. MR 399103, DOI 10.4064/aa-19-3-311-317
Anton Deitmar and Werner Hoffman, Spectral estimates for towers of noncompact quotients, Canad. J. Math. 51 (1999), no. 2, 266–293. MR 1697144, DOI 10.4153/CJM-1999-014-3
J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315, DOI 10.1007/978-3-662-03626-6
Vincent Emery, Torsion homology of arithmetic lattices and $K_2$ of imaginary fields, Math. Z. 277 (2014), no. 3-4, 1155–1164. MR 3229985, DOI 10.1007/s00209-014-1298-2
Tobias Finis, Fritz Grunewald, and Paulo Tirao, The cohomology of lattices in $\textrm {SL}(2,\Bbb C)$, Experiment. Math. 19 (2010), no. 1, 29–63. MR 2649984, DOI 10.1080/10586458.2010.10129067
G. Harder, Eisenstein cohomology of arithmetic groups. The case $\textrm {GL}_2$, Invent. Math. 89 (1987), no. 1, 37–118. MR 892187, DOI 10.1007/BF01404673
Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
Thang T. Q. Lê, Growth of homology torsion in finite coverings and hyperbolic volume, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 2, 611–645 (English, with English and French summaries). MR 3803114, DOI 10.5802/aif.3173
Pere Menal-Ferrer and Joan Porti, Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol. 7 (2014), no. 1, 69–119. MR 3180614, DOI 10.1112/jtopol/jtt024
Simon Marshall and Werner Müller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math. J. 162 (2013), no. 5, 863–888. MR 3047468, DOI 10.1215/00127094-2080850
Werner Müller and Jonathan Pfaff, Analytic torsion of complete hyperbolic manifolds of finite volume, J. Funct. Anal. 263 (2012), no. 9, 2615–2675. MR 2967302, DOI 10.1016/j.jfa.2012.08.020
Werner Müller and Jonathan Pfaff, The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume, J. Funct. Anal. 267 (2014), no. 8, 2731–2786. MR 3255473, DOI 10.1016/j.jfa.2014.08.005
Werner Müller and Jonathan Pfaff, On the growth of torsion in the cohomology of arithmetic groups, Math. Ann. 359 (2014), no. 1-2, 537–555. MR 3201905, DOI 10.1007/s00208-014-1014-x
Jonathan Pfaff, Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups, Ann. Global Anal. Geom. 45 (2014), no. 4, 267–285. MR 3180949, DOI 10.1007/s10455-013-9400-2
Jonathan Pfaff, A gluing formula for the analytic torsion on hyperbolic manifolds with cusps, J. Inst. Math. Jussieu 16 (2017), no. 4, 673–743. MR 3680342, DOI 10.1017/S1474748015000237
J. Raimbault, Analytic, Reidemeister and homological torsion for congruence three-manifolds, arXiv:1307.2845 (2013); Ann. Fac. Sci. Toulouse Math. (6) (to appear).
Roman Sauer, Volume and homology growth of aspherical manifolds, Geom. Topol. 20 (2016), no. 2, 1035–1059. MR 3493098, DOI 10.2140/gt.2016.20.1035
Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), no. 3, 945–1066. MR 3418533, DOI 10.4007/annals.2015.182.3.3