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

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.

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.

Published online:
DOI: 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
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  - 
%0 Journal Article
%T Traces, lengths, axes and commensurability
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2014
%P 1103-1118
%V Ser. 6, 23
%N 5
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1438
%R 10.5802/afst.1438
%G en
%F AFST_2014_6_23_5_1103_0
Alan W. Reid. Traces, lengths, axes and commensurability. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 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

Cited by Sources: