logo AFST
Traces, lengths, axes and commensurability
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 5, pp. 1103-1118.

Cet article est bâtit autour de la question suivante : comment des propriétés géométriques et analytiques de variétés hyperboliques de dimension 3 déterminent-elles leurs classes de commensurabilité. Cet article est pour la plus grande partie un aperçu de travaux récents.

The focus of this paper are questions related to how various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. The paper is for the large part a survey of recent work.

Publié le :
DOI : https://doi.org/10.5802/afst.1438
@article{AFST_2014_6_23_5_1103_0,
     author = {Alan W. Reid},
     title = {Traces, lengths, axes and commensurability},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1103--1118},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {5},
     year = {2014},
     doi = {10.5802/afst.1438},
     mrnumber = {3294604},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1438/}
}
TY  - JOUR
AU  - Alan W. Reid
TI  - Traces, lengths, axes and commensurability
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2014
DA  - 2014///
SP  - 1103
EP  - 1118
VL  - Ser. 6, 23
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1438/
UR  - https://www.ams.org/mathscinet-getitem?mr=3294604
UR  - https://doi.org/10.5802/afst.1438
DO  - 10.5802/afst.1438
LA  - en
ID  - AFST_2014_6_23_5_1103_0
ER  - 
Alan W. Reid. Traces, lengths, axes and commensurability. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 5, pp. 1103-1118. doi : 10.5802/afst.1438. https://afst.centre-mersenne.org/articles/10.5802/afst.1438/

[1] Agol (I.).— The virtual Haken conjecture, with an appendix by I. Agol, D. Groves, and J. Manning, Documenta Math. 18 (2013), p. 1045-1087. | MR 3104553 | Zbl 1286.57019

[2] Brooks (R.).— Constructing isospectral manifolds, Amer. Math. Monthly 95, (1988) p. 823-839. | MR 967343 | Zbl 0673.58046

[3] Brooks (R.).— The Sunada method, in The Tel Aviv Topology Conference: Rothenberg Festschrift (1998), Contemp. Math. 231, p. 25-35 Amer. Math. Soc. (1999). | MR 1705572 | Zbl 0935.58018

[4] Chinburg (T.), Hamilton (E.), Long (D. D.) and Reid (A. W.).— Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds, Duke Math. J. 145 (2008), p. 25-44. | MR 2451288 | Zbl 1169.53030

[5] Conder (M. D. E.).— Hurwitz groups: A brief survey, Bulletin A. M. S. 23, (1990), p. 359-370. | MR 1041434 | Zbl 0716.20015

[6] Gangolli (R.).— The length spectra of some compact manifolds, J. Diff. Geom. 12 (1977), p. 403-424. | MR 650997 | Zbl 0365.53016

[7] Guralnick (R. M.).— Subgroups inducing the same permutation representation, J. Algebra 81 (1983), p. 312-319. | MR 700287 | Zbl 0527.20005

[8] Lakeland (G. S.).— Equivalent trace sets for arithmetic Fuchsian groups, preprint (2013).

[9] Leininger (C.), McReynolds (D. B.), Neumann (W. D.) and Reid (A. W.).— Length and eigenvalue equivalence, International Math. Research Notices 2007, article ID rnm135, 24 pages, | MR 2377017 | Zbl 1158.53032

[10] Long (D. D.) and Reid (A. W.).— On Fuchsian groups with the same set of axes, Bull. London Math. Soc. 30 (1998), p. 533-538. | MR 1643818 | Zbl 0935.20038

[11] Lubotzky (A.), Samuels (B.) and Vishne (U.).— Division algebras and non-commensurable isospectral manifolds, Duke Math. J. 135 (2006), p. 361-379. | MR 2267287 | Zbl 1123.58020

[12] Maclachlan (C.) and Reid (A. W.).— The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics 219, Springer-Verlag (2003). | MR 1937957 | Zbl 1025.57001

[13] McReynolds (D. B.).— Isospectral locally symmetric manifolds, to appear Indiana J. Math. | MR 3233218

[14] Millson (J. J.).— On the first Betti number of a constant negatively curved manifold, Annals of Math. 104 (1976), p. 235-247. | MR 422501 | Zbl 0364.53020

[15] Neumann (W. D.) and Reid (A. W.).— Arithmetic of hyperbolic manifolds, in TOPOLOGY ’90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University. De Gruyter Verlag (1992), p. 273-310. | MR 1184416 | Zbl 0777.57007

[16] Pesce (H.).— Compacité de l’ensemble des réseaux isospectraux et conséquences, Topology 36 (1997), p. 695-710. | MR 1422430 | Zbl 0874.58089

[17] Perlis (R.).— On the equation ζ K (s)=ζ K ' (s), J. Number Theory 9 (1977), p. 342-360. | MR 447188 | Zbl 0389.12006

[18] Prasad (G.) and Rapinchuk (A. S.).— Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. I. H. E. S. 109 (2009), p. 113-184. | Numdam | MR 2511587 | Zbl 1176.22011

[19] Reid (A. W.).— Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds, Duke Math. J. 65 (1992), p. 215-228. | MR 1150584 | Zbl 0776.58040

[20] Reid (A. W.).— The geometry and topology of arithmetic hyperbolic 3-manifolds, Proc. Symposium Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces, Kyoto 2006, RIMS Kokyuroku Series 1571 (2007), p. 31-58.

[21] Schmutz (P.).— Arithmetic groups and the length spectrum of Riemann surfaces, Duke Math. J. 84 (1996), p. 199-215. | MR 1394753 | Zbl 0867.30030

[22] Sunada (T.).— Riemannian coverings and isospectral manifolds, Annals of Math. 121 (1985), p. 169-186. | MR 782558 | Zbl 0585.58047

[23] Takeuchi (K.).— A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), p. 600-612. | MR 398991 | Zbl 0311.20030

[24] Vignéras (M-F.).— Variétiés Riemanniennes isospectrales et non isométriques, Annals of Math. 112 (1980), p. 21-32. | Zbl 0445.53026

Cité par Sources :