The arithmetic hyperbolic 3-manifold of smallest volume

Ted Chinburg; Eduardo Friedman; Kerry N. Jones; Alan W. Reid

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2001)

  • Volume: 30, Issue: 1, page 1-40
  • ISSN: 0391-173X

How to cite

top

Chinburg, Ted, et al. "The arithmetic hyperbolic 3-manifold of smallest volume." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 30.1 (2001): 1-40. <http://eudml.org/doc/84436>.

@article{Chinburg2001,
author = {Chinburg, Ted, Friedman, Eduardo, Jones, Kerry N., Reid, Alan W.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Weeks manifold; Meyerhoff manifold},
language = {eng},
number = {1},
pages = {1-40},
publisher = {Scuola normale superiore},
title = {The arithmetic hyperbolic 3-manifold of smallest volume},
url = {http://eudml.org/doc/84436},
volume = {30},
year = {2001},
}

TY - JOUR
AU - Chinburg, Ted
AU - Friedman, Eduardo
AU - Jones, Kerry N.
AU - Reid, Alan W.
TI - The arithmetic hyperbolic 3-manifold of smallest volume
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2001
PB - Scuola normale superiore
VL - 30
IS - 1
SP - 1
EP - 40
LA - eng
KW - Weeks manifold; Meyerhoff manifold
UR - http://eudml.org/doc/84436
ER -

References

top
  1. [Ad] C. Adams, The noncompact hyperbolic 3-manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987), 601-606. Zbl0634.57008MR894423
  2. [At] M. Atria, Lower bounds for discriminants of octic number fields having six real places, Preprint (1996). 
  3. [BMO] A.M. Bergé - J. Martinet - M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comp. 54 (1990), 869-884. Zbl0709.11056MR1011438
  4. [Bo] A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981), 1-33. Zbl0473.57003MR616899
  5. [Ch] T. Chinburg, A small arithmetic hyperbolic 3-manifold, Proc. Amer. Math. Soc. 100 (1987), 140-144. Zbl0621.57006MR883417
  6. [CF1] T. Chinburg - E. Friedman, The smallest arithmetic hyperbolic three-orbifold, Invent. Math. 86 (1986), 507-527. Zbl0643.57011MR860679
  7. [CF2] T. Chinburg - E. Friedman, Finite subgroups of maximal arithmetic subgroups of PGL(2, C), Ann. Inst. Fourier (Grenoble) 50 (2000), 1765-1798. Zbl0973.20040MR1817383
  8. [CF3] T. Chinburg - E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. (2) 60 (1999), 33-44. Zbl0940.11053MR1721813
  9. [CF4] T. Chinburg - E. Friedman, Hilbert symbols, class groups and quaternion algebras, J. Théorie Nombres Bordeaux12 (2000), 367-377. Zbl0973.11097MR1823190
  10. [Co] H. Cohen et al., PARI, Freeware available by anonymous FTP from megrez.math.u-bordeaux.fr, directory pub/pari. 
  11. [Cu] M. Culler, Lifting representations to covering groups, Adv. Math.59 (1986), 64-70. Zbl0582.57001MR825087
  12. [CS1] M. Culler - P.B. Shalen, Paradoxical decompositions, 2-generator Kleinian groups and volumes of hyperbolic 3-manifolds, J. Amer. Math. Soc.5 (1992), 231-288. Zbl0769.57010MR1135928
  13. [CS2] M. Culler - P.B. Shalen, Volumes of Haken manifolds, I, Invent. Math.118 (1994), 285-329. Zbl0862.57008MR1292114
  14. [CS3] M. Culler - P.B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math.117 (1983), 109-146. Zbl0529.57005MR683804
  15. [CHS] M. Culler - S. Hersonsky - P.B. Shalen, The first betti number of the smallest closed hyperbolic 3-manifold, Topology37 (1998), 805-849. Zbl0893.57015MR1607748
  16. [EEK] A.L. Edmonds - J.H. Ewing - R.S. Kulkarni, Torsion free subgroups of Fuchsian groups and tessellations of surfaces, Invent. Math.69 (1982), 331-346. Zbl0498.20033MR679761
  17. [GMT] D. Gabai - G.R. Meyerhoff - N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic, To appear Ann. of Math. Zbl1052.57019MR1973051
  18. [GMMR] F.W. Gehring - C. Maclachlan - G.J. Martin - A.W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc.349 (1997), 3611-3643. Zbl0889.30031MR1433117
  19. [GM] F.W. Gehring - G.J. Martin, Precisely invariant collars and the volume of hyperbolic 3-folds, J. Differential Geom.49 (1998), 411-435. Zbl0989.57010MR1669657
  20. [Gr] M. Gromov, "Hyperbolic Manifolds According to Thurston and Jørgensen", Séminaire Bourbaki 546, L.N.M. 842, Springer Verlag, Berlin, 1981. Zbl0452.51017MR636516
  21. [HW] C.D. Hodgson - J. Weeks, Symmetries, isometries, and length spectra of closed hyperbolic 3-manifolds, Experimental Math.3 (1994), 101-113. Zbl0841.57020MR1341719
  22. [JRI] K.N. Jones - A.W. Reid, Minimal index torsion-free subgroups of Kleinian groups, Math. Ann.310 (1998), 235-250. Zbl0890.57016MR1602004
  23. [JR2] K.N. Jones - A.W. Reid, Computational methods in arithmetic Kleinian groups, in preparation. 
  24. [MR1] C. Maclachlan - A.W. Reid, Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Cambridge Philos. Soc. 102 (1987), 251-257. Zbl0632.30043MR898145
  25. [MR2] C. Maclachlan - A.W. Reid, "The Arithmetic of Hyperbolic 3-Manifolds ", To appear Springer Verlag. Zbl1025.57001MR1937957
  26. [Ma] J. Martinet, Petits discriminants des corps de nombres, In: "Armitage", J. V. (ed.), Proceedings of the Journeés Arithmétiques1980. London Math. Soc. Lecture Notes series 56Cambridge Univ. Press1982, pp. 151-193. Zbl0491.12005MR697261
  27. [MV] A. Mednykh - A. Vesnin, Visualization of the isometry group action on the Fomenko Matveev Weeks manifold, J. Lie Theory8 (1998), 51-66. Zbl0891.57014MR1616790
  28. [Mo] G.D. Mostow, Quasi-conformal mappings in n-space and the rigidity ofhyperbolic spaceforms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. spaceforms, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 53-104. Zbl0189.09402MR236383
  29. [O1] M. Olivier, Corps sextiques primitifs, Ann. Inst. Fourier (Grenoble) 40 (1990), 757-767. Zbl0734.11054MR1096589
  30. [O2] M. Olivier, The computation of sextic fields with a cubic subfield and no quadratic subfield, Math. Comp. 58 (1992), 419-432. Zbl0746.11041MR1106977
  31. [Po] G. Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, Exposé n° 6Paris, (1976/ 77), 6-01-6-18. Zbl0393.12010MR551335
  32. [Pr] G. Prasad, Q-rank one lattices, Invent. Math.21 (1973), 255-286. Zbl0264.22009MR385005
  33. [Prz] A. Przeworski, Cones embedded in hyperbolic 3-manifolds, Preprint (2000). 
  34. [RW] A.W. Reid - S. Wang, Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree, Comm. Anal. Geom.7 (1999), 105-132. Zbl0930.57012MR1674109
  35. [Th] W. Thurston, "The Geometry and Topology of 3-Manifolds", Mimeographed Lecture Notes, Princeton Univ., 1977. 
  36. [Vi] M.-F. Vignéras, "Arithmétique des algèbres de Quaternions", L.N.M.800, Springer Verlag, Berlin, 1980. Zbl0422.12008MR580949
  37. [We1] J. Weeks, "Hyperbolic Structures on 3-manifolds", Ph. D. Thesis, Princeton University, 1985. 
  38. [We2] J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3- manifolds, available by anonymous ftp from www.northnet.org/weeks/. 

NotesEmbed ?

top

You must be logged in to post comments.