Nicolas Bergeron; Michael Lipnowski
Twisted limit formula for torsion and cyclic base change
(Formule de multiplicité limite tordue pour la torsion et changement de base cyclique)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 435-471, doi: 10.5802/jep.47
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Class. Math.: 11F75, 11F70, 11F72, 58J52
Keywords: Homological torsion, limit multiplicities, base change

Résumé - Abstract

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to $1$, e.g. $G= \mathrm{SL}_2 (\mathbb{C}) \times \mathrm{SL}_2 (\mathbb{C})$ or $\mathrm{SL}_3 (\mathbb{C})$. Then the fundamental rank of $G$ is $2$, and according to the conjecture made in [3], lattices in $G$ should have ‘little’ — in the very weak sense of ‘subexponential in the co-volume’ — torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the square root of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^2$-torsions in the general base-change situation. This also makes uses of a precise equivariant ‘Cheeger-Müller Theorem’ proved by the second author [23].

Bibliography

[1] M. Abert, 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”, arXiv:1210.2961, 2012
[2] D. Barbasch & H. Moscovici, “$L^{2}$-index and the Selberg trace formula”, J. Funct. Anal. 53 (1983) no. 2, p. 151-201 Article
[3] N. Bergeron & A. Venkatesh, “The asymptotic growth of torsion homology for arithmetic groups”, J. Inst. Math. Jussieu 12 (2013) no. 2, p. 391-447 Article
[4] J.-M. Bismut & W. Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205, Société Mathématique de France, Paris, 1992, With an appendix by François Laudenbach
[5] J.-M. Bismut & W. Zhang, “Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle”, Geom. Funct. Anal. 4 (1994) no. 2, p. 136-212 Article
[6] A. Borel, J.-P. Labesse & J. Schwermer, “On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields”, Compositio Math. 102 (1996) no. 1, p. 1-40
[7] A. Bouaziz, “Formule d’inversion d’intégrales orbitales tordues”, Compositio Math. 81 (1992) no. 3, p. 261-290
[8] G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New York-London, 1972  MR 413144
[9] F. Calegari, “Blog post: torsion in the cohomology of co-compact arithmetic lattices”, http://galoisrepresentations.wordpress.com/2013/02/06/
[10] F. Calegari & M. Emerton, “Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms”, Ann. of Math. (2) 170 (2009) no. 3, p. 1437-1446 Article
[11] F. Calegari & M. Emerton, “Mod-$p$ cohomology growth in $p$-adic analytic towers of 3-manifolds”, Groups Geom. Dyn. 5 (2011) no. 2, p. 355-366 Article
[12] F. Calegari & A. Venkatesh, “A torsion Jacquet–Langlands correspondence”, arXiv:1212.3847, 2012
[13] L. Clozel, “Changement de base pour les représentations tempérées des groupes réductifs réels”, Ann. Sci. École Norm. Sup. (4) 15 (1982) no. 1, p. 45-115 Article |  MR 672475
[14] P. Delorme, “Théorème de Paley-Wiener invariant tordu pour le changement de base ${\bf C}/{\bf R}$”, Compositio Math. 80 (1991) no. 2, p. 197-228
[15] Harish-Chandra, “Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula”, Ann. of Math. (2) 104 (1976) no. 1, p. 117-201 Article
[16] R. A. Herb & J. A. Wolf, “The Plancherel theorem for general semisimple groups”, Compositio Math. 57 (1986) no. 3, p. 271-355
[17] S. Illman, “Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group”, Math. Ann. 233 (1978) no. 3, p. 199-220 Article
[18] F. F. Knudsen & D. Mumford, “The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’”, Math. Scand. 39 (1976) no. 1, p. 19-55 Article
[19] J.-P. Labesse, “Pseudo-coefficients très cuspidaux et $K$-théorie”, Math. Ann. 291 (1991) no. 4, p. 607-616 Article |  MR 1135534
[20] J.-P. Labesse & J.-L. Waldspurger, La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series 31, American Mathematical Society, Providence, RI, 2013
[21] R. P. Langlands, Base change for ${\rm GL}(2)$, Annals of Mathematics Studies 96, Princeton University Press, Princeton, N.J., 1980  MR 574808
[22] M. Lipnowski, “Equivariant torsion and base change”, Algebra Number Theory 9 (2015) no. 10, p. 2197-2240 Article
[23] M. Lipnowski, “The equivariant Cheeger–Müller theorem on locally symmetric spaces”, J. Inst. Math. Jussieu 15 (2016) no. 1, p. 165-202, See also arXiv:1312.2543v2; version 2 has corrections that we refer to not appearing in the published JIMJ version Article
[24] J. Lott & M. Rothenberg, “Analytic torsion for group actions”, J. Differential Geom. 34 (1991) no. 2, p. 431-481 Article |  MR 1131439
[25] W. Lück, “Analytic and topological torsion for manifolds with boundary and symmetry”, J. Differential Geom. 37 (1993) no. 2, p. 263-322 Article
[26] W. Müller, “Analytic torsion and $R$-torsion for unimodular representations”, J. Amer. Math. Soc. 6 (1993) no. 3, p. 721-753 Article
[27] W. Müller & J. Pfaff, “On the asymptotics of the Ray-Singer analytic torsion for compact hyperbolic manifolds”, Internat. Math. Res. Notices (2013) no. 13, p. 2945-2983 Article
[28] M. Olbrich, “$L^2$-invariants of locally symmetric spaces”, Doc. Math. 7 (2002), p. 219-237
[29] J. Rohlfs, Lefschetz numbers for arithmetic groups, Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989), Lect. Notes in Math. 1447, Springer, 1990, p. 303–313
[30] J. Rohlfs & B. Speh, “Lefschetz numbers and twisted stabilized orbital integrals”, Math. Ann. 296 (1993) no. 2, p. 191-214 Article
[31] M. H. Şengün, “On the integral cohomology of Bianchi groups”, Experiment. Math. 20 (2011) no. 4, p. 487-505 Article
[32] J.-P. Serre, Cohomologie galoisienne, Lect. Notes in Math. 5, Springer-Verlag, Berlin, 1994  MR 1324577