no. 1
Congruence RFRS towers
[Tours RFRS de congruence]
Agol, Ian1 ; Stover, Matthew2
1 University of California Berkeley, Berkeley, CA (USA)
2 Temple University, Philadelphia, PA (USA)
Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 307-333.

Nous donnons un critère pour qu’un réseau réel ou complexe hyperbolique admette une tour résiduellement finie rationnelle soluble (RFRS) qui se compose entièrement de sous-groupes de congruence. Nous l’utilisons pour montrer que certains groupes de Bianchi PSL 2 (𝒪 d ) sont virtuellement fibrés sur des sous-groupes de congruence, et donnons aussi les premiers exemples de groupes de Kähler RFSR qui ne sont pas des sous-groupes d’un produit de groupes de surface et de groupes abéliens.

We describe a criterion for a real or complex hyperbolic lattice to admit a residually finite rational solvable (RFRS) tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups PSL 2 (𝒪 d ) are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3532
Classification : 20H10, 22E40, 11F06, 20H05, 32Q15, 57K32
Keywords: RFRS towers, Bianchi groups, congruence subgroups, real and complex hyperbolic lattices, virtual fibering, Kähler groups.
Mot clés : Tours RFRS, groupes de Bianchi, sous-groupes de congruence, réseaux réels et complexes hyperboliques, fibre virtuelle, groupes de Kähler.
Agol, Ian 1 ; Stover, Matthew 2

1 University of California Berkeley, Berkeley, CA (USA)
2 Temple University, Philadelphia, PA (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2023__73_1_307_0,
     author = {Agol, Ian and Stover, Matthew},
     title = {Congruence {RFRS} towers},
     journal = {Annales de l'Institut Fourier},
     pages = {307--333},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {1},
     year = {2023},
     doi = {10.5802/aif.3532},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3532/}
}
TY  - JOUR
AU  - Agol, Ian
AU  - Stover, Matthew
TI  - Congruence RFRS towers
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 307
EP  - 333
VL  - 73
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3532/
DO  - 10.5802/aif.3532
LA  - en
ID  - AIF_2023__73_1_307_0
ER  - 
%0 Journal Article
%A Agol, Ian
%A Stover, Matthew
%T Congruence RFRS towers
%J Annales de l'Institut Fourier
%D 2023
%P 307-333
%V 73
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3532/
%R 10.5802/aif.3532
%G en
%F AIF_2023__73_1_307_0
Agol, Ian; Stover, Matthew. Congruence RFRS towers. Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 307-333. doi : 10.5802/aif.3532. https://aif.centre-mersenne.org/articles/10.5802/aif.3532/

[1] Agol, I. Criteria for virtual fibering, J. Topol., Volume 1 (2008) no. 2, pp. 269-284 | DOI | MR | Zbl

[2] Agol, I. The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (with an appendix by Agol, Daniel Groves, and Jason Manning) | MR | Zbl

[3] Agol, I.; Long, D. D.; Reid, A. W. The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. (2), Volume 153 (2001) no. 3, pp. 599-621 | DOI | MR | Zbl

[4] Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y.-S. Smooth compactifications of locally symmetric varieties, Cambridge Mathematical Library, Cambridge University Press, 2010, x+230 pages | DOI | MR | Zbl

[5] Baker, M. D.; Goerner, M.; Reid, A. W. All principal congruence link groups, J. Algebra, Volume 528 (2019), pp. 497-504 | DOI | MR | Zbl

[6] Bergeron, N.; Clozel, L. Spectre automorphe des variétés hyperboliques et applications topologiques, Astérisque, 303, Paris: Société Mathématique de France, 2005, xx+218 pages | MR | Zbl

[7] Bergeron, N.; Clozel, L. Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math., Volume 192 (2013) no. 3, pp. 505-532 | DOI | MR | Zbl

[8] Bergeron, N.; Clozel, L. Sur la cohomologie des variétés hyperboliques de dimension 7 trialitaires, Israel J. Math., Volume 222 (2017) no. 1, pp. 333-400 | DOI | MR | Zbl

[9] Bosma, W.; Cannon, J.; Playoust, C. The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3-4, pp. 235-265 computational algebra and number theory (London, 1993) | DOI | MR | Zbl

[10] Buchstaber, V.; Panov, T. Toric topology, Mathematical Surveys and Monographs, 204, American Mathematical Society, 2015, xiv+518 pages | DOI | MR | Zbl

[11] Chu, Michelle Special subgroups of Bianchi groups, Groups Geom. Dyn., Volume 14 (2020) no. 1, pp. 41-59 | DOI | MR | Zbl

[12] Deligne, P.; Mostow, G. D. Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. (1986) no. 63, pp. 5-89 | DOI | MR | Zbl

[13] Di Cerbo, L. F.; Stover, M. Bielliptic ball quotient compactifications and lattices in PU (2,1) with finitely generated commutator subgroup, Ann. Inst. Fourier (Grenoble), Volume 67 (2017) no. 1, pp. 315-328 http://aif.cedram.org/item?id=AIF_2017__67_1_315_0 | DOI | MR | Zbl

[14] Finis, T.; Grunewald, F.; Tirao, P. The cohomology of lattices in SL(2,ℂ), Experiment. Math., Volume 19 (2010) no. 1, pp. 29-63 http://projecteuclid.org/euclid.em/1268404802 | DOI | MR | Zbl

[15] Friedl, S.; Vidussi, S. Virtual Algebraic Fibrations of Kähler Groups, Nagoya Math. J., Volume 243 (2021), pp. 42-60 | DOI | MR | Zbl

[16] Groves, D.; Manning, J. Quasiconvexity and Dehn filling, Amer. J. Math., Volume 143 (2021) no. 1, pp. 95-124 | DOI | MR | Zbl

[17] Jankiewicz, K.; Norin, S.; Wise, D. Virtually fibering right-angled Coxeter groups, J. Inst. Math. Jussieu, Volume 20 (2021) no. 3, pp. 957-987 | DOI | MR | Zbl

[18] Kielak, D. Residually finite rationally solvable groups and virtual fibring, J. Amer. Math. Soc., Volume 33 (2020) no. 2, pp. 451-486 | DOI | MR | Zbl

[19] Kirwan, F. C.; Lee, R.; Weintraub, S. H. Quotients of the complex ball by discrete groups, Pacific J. Math., Volume 130 (1987) no. 1, pp. 115-141 http://projecteuclid.org/euclid.pjm/1102690295 | DOI | MR | Zbl

[20] Lang, S. Algebraic number theory, Graduate Texts in Mathematics, 110, Springer-Verlag, 1994, xiv+357 pages | DOI | MR | Zbl

[21] Lott, J. Deficiencies of lattice subgroups of Lie groups, Bull. London Math. Soc., Volume 31 (1999) no. 2, pp. 191-195 | DOI | MR | Zbl

[22] Ratcliffe, J. G.; Tschantz, S. T. The volume spectrum of hyperbolic 4-manifolds, Experiment. Math., Volume 9 (2000) no. 1, pp. 101-125 http://projecteuclid.org/euclid.em/1046889595 | DOI | MR | Zbl

[23] Rogawski, J. D. Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, 123, Princeton University Press, 1990 | DOI | MR | Zbl

[24] Schwermer, J. Geometric cycles, arithmetic groups and their cohomology, Bull. Amer. Math. Soc. (N.S.), Volume 47 (2010) no. 2, pp. 187-279 | DOI | MR | Zbl

[25] Serre, Jean-Pierre Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, x+142 pages | MR | Zbl

[26] Stover, M. Cusp and b 1 growth for ball quotients and maps onto ℤ with finitely generated kernel (to appear in Indiana Univ. Math. J.)

[27] Thurston, W. P. Geometry and topology of three-manifolds (unpublished notes: http://library.msri.org/books/gt3m/)

[28] Tits, J. Reductive groups over local fields, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., 1979, pp. 29-69 | MR | Zbl

[29] Wise, D. The structure of groups with a quasiconvex hierarchy, Annals of Mathematics Studies, 209, Princeton University Press, Princeton, NJ, 2021, x+357 pages | MR

[30] Yamazaki, T.; Yoshida, M. On Hirzebruch’s examples of surfaces with c 1 2 =3c 2 , Math. Ann., Volume 266 (1984) no. 4, pp. 421-431 | DOI | MR | Zbl

Cité par Sources :