A.5
Une idée de la preuve des théorèmes A.7 et A.8
Le théorème A.8 est une conséquence immédiate du résultat suivant [Rai12a, Proposition 2.1] et d’un dénombrement trivial.
Proposition A.11. Soient α, β ∈ {0, 1}Z/m. Les variétés hyperboliques M
α1,...,αm et Mβ1,...,βm
sont commensurables si et seulement s’il existe un p ∈ Z/m tel que l’on ait ∀j, αj+p = βj ou
bien ∀j, αp−j = βj.
Le principal ingrédient de la preuve de cette proposition est le lemme suivant, qui est prouvé en utilisant des arguments inspirés de [GPS88].
Lemme A.12. Si W est une variété hyperbolique complète, N00, N10 sont des revêtements finis de N0, N1 respectivement et ι0, ι1 des plongements N00, N10 ,→ W alors ι0(N00)∩ι1(N10) est d’intérieur
vide.
Le théorème A.7 est nettement plus dur à prouver. Cependant il est assez facile de montrer que Nα n’admet aucune isométrie d’ordre infini (et donc qu’elle n’a pas de quotient Galoisien) en utilisant le lemme A.12 et la proposition A.11 (cf. [ABB+, Corollary 13.12]). Le cas général utilise un argument semblable (mais plus compliqué) une fois qu’on a démontré (cf. [ABB+, Lemma 13.17]) :
Lemme A.13. Si Nα a un quotient compact il existe un revêtement fini de Nα qui admette une
isométrie d’ordre infini.
La preuve de ce dernier lemme utilise des résultats profonds et très récents de I. Agol et D. Wise (cf. [ABB+, Proposition 13.14]).
Bibliographie
[ABB+] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault
et I. Samet : On the growth of L2 invariants for sequences of lattices in Lie groups.
In preparation.
[ABB+11] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Ni- kolov, Jean Raimbault et Iddo Samet : On the growth of Betti numbers of locally symmetric spaces. C. R. Math. Acad. Sci. Paris, 349(15-16):831–835, 2011.
[AGM12] I. Agol, D. Groves et J. Manning : The virtual Haken conjecture. ArXiv e-prints, avril 2012.
[Art96] Emil Artin : Algèbre géométrique. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Paris, french édition, 1996. Translated from the 1957 English original by M. Lazard, Edited and with a foreword by G. Julia.
[BC05] Nicolas Bergeron et Laurent Clozel : Spectre automorphe des variétés hyperbo- liques et applications topologiques. Astérisque, (303):xx+218, 2005.
[BD] Jeff Brock et Nathan M. Dunfield : Injectivity radii of integer homology 3–spheres.
En préparation.
[BE06] Nigel Boston et Jordan S. Ellenberg : Pro-p groups and towers of rational ho- mology spheres. Geom. Topol., 10:331–334 (electronic), 2006.
[Bel07] Mikhail Belolipetsky : Counting maximal arithmetic subgroups. Duke Math. J., 140(1):1–33, 2007. With an appendix by Jordan Ellenberg and Akshay Venkatesh. [Ber08] Tobias Berger : Denominators of Eisenstein cohomology classes for GL2 over ima-
ginary quadratic fields. Manuscripta Math., 125(4):427–470, 2008.
[Ber11] Nicolas Bergeron : Le spectre des surfaces hyperboliques. Savoirs actuels. CNRS Editions, EDP sciences, 2011.
[BGLM02] M. Burger, T. Gelander, A. Lubotzky et S. Mozes : Counting hyperbolic manifolds. Geom. Funct. Anal., 12(6):1161–1173, 2002.
[BM11] Jochen Brüning et Xiaonan Ma : On the gluing formula for the analytic torsion.
preprint, 2011.
[BMP03] Michel Boileau, Sylvain Maillot et Joan Porti : Three-dimensional orbifolds and
their geometric structures, volume 15 de Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris, 2003.
[Bow12] L. Bowen : Invariant random subgroups of the free group. ArXiv e-prints, avril 2012.
[BP92] Riccardo Benedetti et Carlo Petronio : Lectures on hyperbolic geometry. Uni- versitext. Springer-Verlag, Berlin, 1992.
[Bre97] Glen E. Bredon : Sheaf theory, volume 170 de Graduate Texts in Mathematics. Springer-Verlag, New York, second édition, 1997.
[Bro82] Kenneth S. Brown : Cohomology of groups, volume 87 de Graduate Texts in Ma-
thematics. Springer-Verlag, New York, 1982.
[BS01] Itai Benjamini et Oded Schramm : Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001.
[Bum97] Daniel Bump : Automorphic forms and representations, volume 55 de Cambridge
Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.
[BV] Nicolas Bergeron et Akshay Venkatesh : The asymptotic growth of torsion homology for arithmetic groups. To appear in JIMJ. math.NT/1004.1083v1. [BW00] A. Borel et N. Wallach : Continuous cohomology, discrete subgroups, and repre-
sentations of reductive groups, volume 67 de Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, second édition, 2000.
[BZ92] Jean-Michel Bismut et Weiping Zhang : An extension of a theorem by Cheeger and Müller. Astérisque, (205):235, 1992. With an appendix by François Laudenbach. [CD06] Frank Calegari et Nathan M. Dunfield : Automorphic forms and rational homo-
logy 3-spheres. Geom. Topol., 10:295–329 (electronic), 2006.
[CdV81] Yves Colin de Verdière : Une nouvelle démonstration du prolongement méro- morphe des séries d’Eisenstein. C. R. Acad. Sci. Paris Sér. I Math., 293(7):361–363, 1981.
[CFK11] Jae Choon Cha, Stefan Friedl et Taehee Kim : The cobordism group of homology cylinders. Compos. Math., 147(3):914–942, 2011.
[Che73] Paul R. Chernoff : Essential self-adjointness of powers of generators of hyperbolic equations. J. Functional Analysis, 12:401–414, 1973.
[Che79] Jeff Cheeger : Analytic torsion and the heat equation. Ann. of Math. (2), 109(2): 259–322, 1979.
[Cul86] Marc Culler : Lifting representations to covering groups. Adv. in Math., 59(1):64– 70, 1986.
[CV10] Frank Calegari et Akshay Venkatesh : A torsion Jacquet-Langlands conjecture.
preprint, 2010.
[CW03] Bryan Clair et Kevin Whyte : Growth of Betti numbers. Topology, 42(5):1125– 1142, 2003.
[dGW78] David L. de George et Nolan R. Wallach : Limit formulas for multiplicities in
L2(Γ\G). Ann. of Math. (2), 107(1):133–150, 1978.
[DLM+03] Józef Dodziuk, Peter Linnell, Varghese Mathai, Thomas Schick et Stuart Yates : Approximating L2-invariants and the Atiyah conjecture. Comm. Pure Appl. Math., 56(7):839–873, 2003. Dedicated to the memory of Jürgen K. Moser.
[Don79] Harold Donnelly : Asymptotic expansions for the compact quotients of properly discontinuous group actions. Illinois J. Math., 23(3):485–496, 1979.
[DT06] Nathan M. Dunfield et William P. Thurston : Finite covers of random 3- manifolds. Invent. Math., 166(3):457–521, 2006.
[EGM98] J. Elstrodt, F. Grunewald et J. Mennicke : Groups acting on hyperbolic space. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. Harmonic ana- lysis and number theory.
Bibliographie 175
[Eme12] Vincent Emery : Arbitrarily large families of spaces with the same volume. Geome-
triae dedicata, 156(1), 2012.
[EW99] Graham Everest et Thomas Ward : Heights of polynomials and entropy in alge-
braic dynamics. Universitext. Springer-Verlag London Ltd., London, 1999.
[FH91] William Fulton et Joe Harris : Representation theory, volume 129 de Graduate
Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in
Mathematics.
[FLM12] T. Finis, E. Lapid et W. Mueller : Limit multiplicities for principal congruence subgroups of GL(n). ArXiv e-prints, août 2012.
[GAS91] Francisco González-Acuña et Hamish Short : Cyclic branched coverings of knots and homology spheres. Rev. Mat. Univ. Complut. Madrid, 4(1):97–120, 1991. [Gil95] Peter B. Gilkey : Invariance theory, the heat equation, and the Atiyah-Singer index
theorem. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second
édition, 1995.
[God66] R. Godement : The decomposition of L2(G/Γ) for Γ = SL(2, Z). In Algebraic
Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo,
pages 211–224. Amer. Math. Soc., Providence, R.I., 1966.
[Gor72] C. McA. Gordon : Knots whose branched cyclic coverings have periodic homology.
Trans. Amer. Math. Soc., 168:357–370, 1972.
[GPS88] M. Gromov et I. Piatetski-Shapiro : Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math., (66):93–103, 1988.
[Har87] G. Harder : Eisenstein cohomology of arithmetic groups. The case GL2. Invent.
Math., 89(1):37–118, 1987.
[Hat02] Allen Hatcher : Algebraic topology. Cambridge University Press, Cambridge, 2002. [Hem76] John Hempel : 3-Manifolds. Princeton University Press, Princeton, N. J., 1976.
Ann. of Math. Studies, No. 86.
[Hum80] James E. Humphreys : Arithmetic groups, volume 789 de Lecture Notes in Mathe-
matics. Springer, Berlin, 1980.
[Kna97] A. W. Knapp : Theoretical aspects of the trace formula for GL(2). In Representation
theory and automorphic forms (Edinburgh, 1996), volume 61 de Proc. Sympos. Pure Math., pages 355–405. Amer. Math. Soc., Providence, RI, 1997.
[Kna01] Anthony W. Knapp : Representation theory of semisimple groups. Princeton Land- marks in Mathematics. Princeton University Press, Princeton, NJ, 2001. An overview based on examples, Reprint of the 1986 original.
[Kob12] Thomas Koberda : Homological eigenvalues of mapping classes and torsion homo- logy growth for fibered 3–manifolds. preprint, 2012.
[Kow08] Emmanuel Kowalski : The large sieve and its applications, volume 175 de Cam-
bridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2008. Arith-
metic geometry, random walks and discrete groups.
[Lam06] T. Y. Lam : Serre’s problem on projective modules. Springer Monographs in Mathe- matics. Springer-Verlag, Berlin, 2006.
[Law83] Wayne M. Lawton : A problem of Boyd concerning geometric means of polynomials.
J. Number Theory, 16(3):356–362, 1983.
[Lic97] W. B. Raymond Lickorish : An introduction to knot theory, volume 175 de Graduate
Texts in Mathematics. Springer-Verlag, New York, 1997.
[LLS11] Peter Linnell, Wolfgang Lück et Roman Sauer : The limit of Fp-Betti numbers of a tower of finite covers with amenable fundamental groups. Proc. Amer. Math.
Soc., 139(2):421–434, 2011.
[LS99] W. Lück et T. Schick : L2-torsion of hyperbolic manifolds of finite volume. Geom.
Funct. Anal., 9(3):518–567, 1999.
[LS03] Alexander Lubotzky et Dan Segal : Subgroup growth, volume 212 de Progress in
Mathematics. Birkhäuser Verlag, Basel, 2003.
[LT12] Hanfeng Li et Andreas Thom : Entropy, determinants and L2-torsion. ArXiv e-
prints, février 2012.
[Lüc94a] W. Lück : Approximating L2-invariants by their finite-dimensional analogues.
Geom. Funct. Anal., 4(4):455–481, 1994.
[Lüc94b] Wolfgang Lück : L2-torsion and 3-manifolds. In Low-dimensional topology (Knox-
ville, TN, 1992), Conf. Proc. Lecture Notes Geom. Topology, III, pages 75–107. Int.
Press, Cambridge, MA, 1994.
[Lüc02] Wolfgang Lück : L2-invariants : theory and applications to geometry and K-theory,
volume 44 de Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer- Verlag, Berlin, 2002.
[LZ06] Weiping Li et Weiping Zhang : An L2-Alexander invariant for knots. Commun.
Contemp. Math., 8(2):167–187, 2006.
[MM82] John P. Mayberry et Kunio Murasugi : Torsion-groups of abelian coverings of links. Trans. Amer. Math. Soc., 271(1):143–173, 1982.
[MP10] P. Menal-Ferrer et J. Porti : Twisted cohomology for hyperbolic three manifolds.
ArXiv e-prints, janvier 2010.
[MP11] P. Menal-Ferrer et J. Porti : Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. ArXiv e-prints, octobre 2011.
[MP12] Werner Müller et Jonathan Pfaff : Analytic torsion of complete hyperbolic ma- nifolds of finite volume. J. Funct. Anal., 263(9):2615–2675, 2012.
[MR03] Colin Maclachlan et Alan W. Reid : The arithmetic of hyperbolic 3-manifolds, volume 219 de Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. [Mül78] Werner Müller : Analytic torsion and R-torsion of Riemannian manifolds. Adv. in
Math., 28(3):233–305, 1978.
[Mül93] Werner Müller : Analytic torsion and R-torsion for unimodular representations.
J. Amer. Math. Soc., 6(3):721–753, 1993.
[Nar04] Władysław Narkiewicz : Elementary and analytic theory of algebraic numbers. Springer Monographs in Mathematics. Springer-Verlag, Berlin, third édition, 2004. [Par09] Jinsung Park : Analytic torsion and Ruelle zeta functions for hyperbolic manifolds
with cusps. J. Funct. Anal., 257(6):1713–1758, 2009.
[Ped97] Emmanuel Pedon : Analyse des formes différentielles sur l’espace hyperbolique réel. Thèse de doctorat, Univerité Henri Poicaré (Nancy 1), 1997.
[Pfa12] J. Pfaff : Selberg zeta functions on odd-dimensional hyperbolic manifolds of finite volume. ArXiv e-prints, mai 2012.
Bibliographie 177
[Por04] Joan Porti : Mayberry-Murasugi’s formula for links in homology 3-spheres. Proc.
Amer. Math. Soc., 132(11):3423–3431 (electronic), 2004.
[Rai12a] Jean Raimbault : A note on maximal subgroup growth in SO(1, n). preprint, à
paraître dans IMRN, 2012.
[Rai12b] Jean Raimbault : Exponential growth of torsion in abelian coverings. Algebr. Geom.
Topol., 12(3):1331–1372, 2012.
[Ril90] Robert Riley : Growth of order of homology of cyclic branched covers of knots.
Bull. London Math. Soc., 22(3):287–297, 1990.
[RS71] D. B. Ray et I. M. Singer : R-torsion and the Laplacian on Riemannian manifolds.
Advances in Math., 7:145–210, 1971.
[Sav89] Gordan Savin : Limit multiplicities of cusp forms. Invent. Math., 95(1):149–159, 1989.
[Sch00] A. Schinzel : Polynomials with special regard to reducibility, volume 77 de Encyclo-
pedia of Mathematics and its Applications. Cambridge University Press, Cambridge,
2000. With an appendix by Umberto Zannier.
[Şen11] Mehmet Haluk Şengün : On the integral cohomology of Bianchi groups. Exp. Math., 20(4):487–505, 2011.
[Ser70] Jean-Pierre Serre : Le problème des groupes de congruence pour SL2. Ann. of
Math. (2), 92:489–527, 1970.
[Ser00] Jean-Pierre Serre : Local algebra. Springer Monographs in Mathematics. Springer- Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author.
[Sie89] Carl Ludwig Siegel : Lectures on the geometry of numbers. Springer-Verlag, Berlin, 1989. Notes by B. Friedman, Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter, With a preface by Chandrasekharan.
[SW02a] Daniel S. Silver et Susan G. Williams : Mahler measure, links and homology growth. Topology, 41(5):979–991, 2002.
[SW02b] Daniel S. Silver et Susan G. Williams : Torsion numbers of augmented groups with applications to knots and links. Enseign. Math. (2), 48(3-4):317–343, 2002. [SX91] Peter Sarnak et Xiao Xi Xue : Bounds for multiplicities of automorphic represen-
tations. Duke Math. J., 64(1):207–227, 1991.
[Tur01] Vladimir Turaev : Introduction to combinatorial torsions. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2001. Notes taken by Felix Schlenk.
[Wan72] Hsien Chung Wang : Topics on totally discontinuous groups. In Symmetric spaces
(Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), pages 459–487. Pure
and Appl. Math., Vol. 8. Dekker, New York, 1972.
[War79] Garth Warner : Selberg’s trace formula for nonuniform lattices : the R-rank one case. In Studies in algebra and number theory, volume 6 de Adv. in Math. Suppl.
Stud., pages 1–142. Academic Press, New York, 1979.
[Web79] Claude Weber : Sur une formule de R. H. Fox concernant l’homologie des revête- ments cycliques. Enseign. Math. (2), 25(3-4):261–272 (1980), 1979.
[Zim94] B. Zimmermann : A note on hyperbolic 3-manifolds of the same volume. Monatsh.