×

Arithmetic hyperbolic surface bundles. (English) Zbl 0830.57008

The authors consider which finite-volume hyperbolic 3-manifolds fibering over the circle are arithmetic. They show, in the non-compact case, that for any given topological type of fibre, there are only finitely many arithmetic examples up to cyclic commensurability. In the compact case, this is also true, given, in addition, a bound on the degree of the trace field – this additional constraint being conjectured superfluous. When the fibre is a once-punctured torus, they give a complete classification: namely, the arithmetic manifolds are precisely the cyclic covers, or sisters of cyclic covers, of one of the three standard examples – those having monodromies with left-right decompositions respectively \(LR\), \(L^2 R\) and \(L^2 R^2\).

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
11G35 Varieties over global fields
22E40 Discrete subgroups of Lie groups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] H. Bass. Groups of integral representation type, Pacific J. Math.86 (1980), 15-51 · Zbl 0444.20006
[2] A.F. Beardon. The Geometry of Discrete Groups, Graduate Texts in Math.91 (Springer Verlag) 1983
[3] S. Betley, J. Przytycki and T. Zukowski. Hyperbolic structures on Dehn filling of some punctured torus bundles overS 1. Kobe J. Math.3 (1987) 117-147
[4] J. Blume-Nienhaus. Doctoral dissertation. Bonn (1991)
[5] F. Bonahon. Bouts des vari?t?s hyperboliques de dimension 3, Annals of Math.124 (1986) 71-158 · Zbl 0671.57008 · doi:10.2307/1971388
[6] A. Borel. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa8 (1981), 1-33 · Zbl 0473.57003
[7] B.H. Bowditch. A proof of McShane’s identity via Markoff triples. To appear in Bull. London Math. Soc. · Zbl 0854.57009
[8] B.H. Bowditch. A variation of McShane’s identity for once-punctured torus bundles. Preprint, Southampton (1994) · Zbl 0872.57015
[9] B.H. Bowditch. Markoff triples and quasifuchsian groups. Preprint, Southampton (1994) · Zbl 0928.11030
[10] A. Casson and S. Bleiler. Automorphisms of surfaces, after Neilsen and Thurston. Cambridge University Press (1988) · Zbl 0649.57008
[11] D. Cooper, D.D. Long and A.W. Reid. Bundles and finite foliations. Invent. Math.118 (1994) 255-283 · Zbl 0858.57015 · doi:10.1007/BF01231534
[12] M. Culler. Lifting representations to covering groups: Adv. in Math.59 (1986) 64-70 · Zbl 0582.57001 · doi:10.1016/0001-8708(86)90037-X
[13] M. Culler, W. Jaco and H. Rubinstein. Incompressible surfaces in once punctured torus bundles. Proc London Math Soc45 (1982) 385-419 · Zbl 0515.57002 · doi:10.1112/plms/s3-45.3.385
[14] E. Dobrowolski. On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith.34 (1979) 391-401 · Zbl 0416.12001
[15] A. Fathi, F. Laudenbach and V. Poenaru. Travaux de Thurston sur les surfaces. Aste?isque 66-67 (1979)
[16] W. Floyd and A. Hatcher. Incompressible surfaces in punctured torus bundles. Topology and its Appl.13 (1982) 263-282 · Zbl 0493.57004 · doi:10.1016/0166-8641(82)90035-9
[17] F. Grunewald and U. Hirsch. Link complements arising from arithmetic group actions. Preprint · Zbl 0840.57007
[18] F. Grunewald and J. Schwermer. Arithmetic quotients of hyperbolic 3-space, cusp forms and link complements. Duke Math J.48 (1981) 351-358 · Zbl 0485.57005 · doi:10.1215/S0012-7094-81-04820-1
[19] W. Jaco, Lectures on 3-Manifolds, CBMS43 (1977)
[20] T. J?rgensen. Compact 3-manifolds of constant negative curvature fibering over the circle, Annals of Math.106 (1977) 61-72 · Zbl 0368.53025 · doi:10.2307/1971158
[21] T. J?rgensen. On pairs of once punctured tori. Unpublished manuscript
[22] T. J?rgensen and A Marden. Two doubly generated groups. Quart J of Maths. Oxford (2)30 (1979) 143-156 · Zbl 0414.30035 · doi:10.1093/qmath/30.2.143
[23] G. Kern-Isberner and G. Rosenberger. Einige Bemerkung ?ber Untergruppen derPSL(2,?). Resultate der Mathematik.6 (1983) 40-47 · Zbl 0517.20023
[24] C. Maclachlan. Fuchsian subgroups of the groupsPSL(2,O d ). In Low-dimensional Topology and Kleinian Groups, Ed. D.B.A. Epstein, London Math. Soc. Lecture Series112, 305-311. Cambridge University Press 1986
[25] C. Maclachlan and A.W. Reid. Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Camb. Phil. Soc.102 (1987) 251-257 · Zbl 0632.30043 · doi:10.1017/S030500410006727X
[26] D. McCullough. Automorphisms of punctured surface bundles. In Geometry and Topology, Ed. C. McCrory. Marcel-Dekker (1987) 179-209 · Zbl 0607.57009
[27] W.D. Neumann and A.W. Reid. Arithmetic of hyperbolic 3-manifolds. In TOPOLOGY ’90, Proceedings of the Research Semester in Low-Dimensional Topology at Ohio State University, 273-310. Editors, B. Apanasov, W.D. Neumann, A.W. Reid and L. Siebenmann, De Gruyter Verlag 1992 · Zbl 0777.57007
[28] A.W. Reid. Ph.D Thesis, University of Aberdeen, 1987
[29] A.W. Reid. A note on trace-fields of Kleinian groups. Bull. London Math. Soc.22 (1990) 349-352 · Zbl 0706.20038 · doi:10.1112/blms/22.4.349
[30] A.W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2)43 (1991) 171-184 · Zbl 0847.57013 · doi:10.1112/jlms/s2-43.1.171
[31] A.W. Reid. Isospectrality and commensurability of arithmetic hyperbolic 2 and 3-manifolds. Duke Math. J.65 (1992) 215-228 · Zbl 0776.58040 · doi:10.1215/S0012-7094-92-06508-2
[32] A.W. Reid. A non-Haken hyperbolic 3-manifold covered by a surface bundle. To appear in Pacific J. Math. · Zbl 0817.57014
[33] R. Riley. A quadratic parabolic group. Math. Proc. Camb. Phil. Soc.77 (1975) 281-288 · Zbl 0309.55002 · doi:10.1017/S0305004100051094
[34] D. Rolfsen. Knots and Links. Publish or Perish 1976 · Zbl 0339.55004
[35] G. Rosenberger. Some remarks on a paper of A.F. Beardon and P.L. Waterman about strongly discrete subgroups ofSL(2,?). J. London Math Soc.27 (1983) 39-42 · Zbl 0495.30038 · doi:10.1112/jlms/s2-27.1.39
[36] G. Rosenberger. Simultaneous conjugation inSL(2,?). Notes
[37] G. Rosenberger and W. Plesken. Simultaneous conjugation in quaternion algebra. Results in Maths.25 (1994) 120-124 · Zbl 0799.20043
[38] C Series. The geometry of Markoff numbers. Math Intelligencer7 (1985) 20-29 · Zbl 0566.10024 · doi:10.1007/BF03025802
[39] J. Silverman. The Markoff equationX 2+Y 2+Z 2=aXYZ over quadratic imaginary fields. J of Number Theory35 (1990) 72-104 · Zbl 0702.11012 · doi:10.1016/0022-314X(90)90105-Z
[40] C. Smyth. On products of conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc.3 (1971) 169-175 · Zbl 0235.12003 · doi:10.1112/blms/3.2.169
[41] T. Soma. Virtual fibers in hyperbolic 3-manifolds. Topology and its Appl.41 (1991) 179-192 · Zbl 0753.57011 · doi:10.1016/0166-8641(91)90002-4
[42] T. Soma. Virtual fiber groups in 3-manifold groups. J. London Math. Soc (2)43 (1991) 337-354 · Zbl 0743.57010 · doi:10.1112/jlms/s2-43.2.337
[43] K. Takeuchi. Arithmetic Fuchsian groups of signature (1;e). J. Math. Soc. Japan35 (1981) 381-407 · Zbl 0517.20022 · doi:10.2969/jmsj/03530381
[44] W.P. Thurston. The Geometry and Topology of 3-Manifolds, Mimeographed lecture notes, Princeton University, 1977
[45] W.P. Thurston. Hyperbolic structures on 3-manifolds II: surface groups and 3-manifolds that fiber of the circle. To appear Annals of Math.
[46] W.P. Thurston. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. A.M.S.6 (1982), 357-381 · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0
[47] M-F. Vign?ras. Arithm?tique des alg?bres de quaternions. L.N.M 800, Springer-Verlag 1980
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.