Vol3 and other exceptional hyperbolic 3manifolds
Authors:
K. N. Jones and A. W. Reid
Journal:
Proc. Amer. Math. Soc. 129 (2001), 21752185
MSC (2000):
Primary 57M50
Published electronically:
December 4, 2000
MathSciNet review:
1825931
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.
 1.
Hyman
Bass, Groups of integral representation type, Pacific J. Math.
86 (1980), no. 1, 15–51. MR 586867
(82c:20014)
 2.
A.
Borel, Commensurability classes and volumes of hyperbolic
3manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
8 (1981), no. 1, 1–33. MR 616899
(82j:22008)
 3.
Ted
Chinburg and Eduardo
Friedman, The smallest arithmetic hyperbolic threeorbifold,
Invent. Math. 86 (1986), no. 3, 507–527. MR 860679
(88a:22022), http://dx.doi.org/10.1007/BF01389265
 4.
T. Chinburg, E. Friedman, K. N. Jones and A. W. Reid, The arithmetic hyperbolic 3manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa, to appear.
 5.
David
Gabai, On the geometric and topological
rigidity of hyperbolic 3manifolds, J. Amer.
Math. Soc. 10 (1997), no. 1, 37–74. MR 1354958
(97h:57028), http://dx.doi.org/10.1090/S0894034797002063
 6.
D. Gabai, R. Meyerhoff and N. Thurston, Homotopy hyperbolic 3manifolds are hyperbolic, Preprint.
 7.
Hugh
M. Hilden, María
Teresa Lozano, and José
María MontesinosAmilibia, A characterization of arithmetic
subgroups of 𝑆𝐿(2,𝑅) and
𝑆𝐿(2,𝐶), Math. Nachr. 159
(1992), 245–270. MR 1237113
(94i:20088), http://dx.doi.org/10.1002/mana.19921590117
 8.
K. N. Jones and A. W. Reid, Computational methods in arithmetic Kleinian groups, in preparation.
 9.
C.
Maclachlan and A.
W. Reid, Commensurability classes of arithmetic Kleinian groups and
their Fuchsian subgroups, Math. Proc. Cambridge Philos. Soc.
102 (1987), no. 2, 251–257. MR 898145
(88j:20040), http://dx.doi.org/10.1017/S030500410006727X
 10.
Walter
D. Neumann and Alan
W. Reid, Arithmetic of hyperbolic manifolds, Topology
’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ.,
vol. 1, de Gruyter, Berlin, 1992, pp. 273–310. MR 1184416
(94c:57024)
 11.
Alan
W. Reid, A note on tracefields of Kleinian groups, Bull.
London Math. Soc. 22 (1990), no. 4, 349–352. MR 1058310
(91d:20056), http://dx.doi.org/10.1112/blms/22.4.349
 12.
Alan
W. Reid, A nonHaken hyperbolic 3manifold covered by a surface
bundle, Pacific J. Math. 167 (1995), no. 1,
163–182. MR 1318168
(95m:57025)
 13.
MarieFrance
Vignéras, Arithmétique des algèbres de
quaternions, Lecture Notes in Mathematics, vol. 800, Springer,
Berlin, 1980 (French). MR 580949
(82i:12016)
 1.
 H. Bass, Groups of integral representation type, Pacific J. Math. 86 (1980), 1551. MR 82c:20014
 2.
 A. Borel, Commensurability classes and volumes of hyperbolic 3manifolds, Ann. Scuola Norm. Sup. Pisa 8, (1981), 133. MR 82j:22008
 3.
 T. Chinburg and E. Friedman, The smallest arithmetic hyperbolic 3orbifold, Invent. Math. 86, (1980), 507527. MR 88a:22022
 4.
 T. Chinburg, E. Friedman, K. N. Jones and A. W. Reid, The arithmetic hyperbolic 3manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa, to appear.
 5.
 D. Gabai, The geometric and topological rigidity of hyperbolic 3manifolds, J. A. M. S. 10 (1997), 3774. MR 97h:57028
 6.
 D. Gabai, R. Meyerhoff and N. Thurston, Homotopy hyperbolic 3manifolds are hyperbolic, Preprint.
 7.
 H. M. Hilden, M.T. Lozano and J.M. Montesinos, A characterization of arithmetic subgroups of and , Math. Nachr. 159 (1992), 245270. MR 94i:20088
 8.
 K. N. Jones and A. W. Reid, Computational methods in arithmetic Kleinian groups, in preparation.
 9.
 C. Maclachlan and A. W. Reid, Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups, Math. Proc. Camb. Phil. Soc. 102 (1987), 251257. MR 88j:20040
 10.
 W. D. Neumann and A. W. Reid, Arithmetic of hyperbolic 3Manifolds, Topology '90, Proc. of Lowdimensional Topology Conference, Ohio State Univ., De Gruyter (1991), 273310. MR 94c:57024
 11.
 A. W. Reid, A note on the tracefields of Kleinian groups, Bull. L. M. S. 22 (1990), 349352. MR 91d:20056
 12.
 A. W. Reid, A nonHaken hyperbolic 3manifold covered by a surface bundle, Pacific J. Math. 167 (1995), 163182. MR 95m:57025
 13.
 M.F. Vignéras, Arithmétique des algèbres de Quaternions, L.N.M. 800, Springer Verlag, Berlin, 1980. MR 82i:12016
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
57M50
Retrieve articles in all journals
with MSC (2000):
57M50
Additional Information
K. N. Jones
Affiliation:
Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47304
Email:
kerryj@math.bsu.edu
A. W. Reid
Affiliation:
Department of Mathematics, University of Texas, Austin, Texas 78712
Email:
areid@math.utexas.edu
DOI:
http://dx.doi.org/10.1090/S0002993900057750
PII:
S 00029939(00)057750
Keywords:
Hyperbolic 3manifold,
arithmetic manifold,
homotopy hyperbolic 3manifold
Received by editor(s):
April 19, 1999
Received by editor(s) in revised form:
October 13, 1999, and November 8, 1999
Published electronically:
December 4, 2000
Additional Notes:
The first author was partially supported by Ball State University.
The second author was partially supported by the Royal Society, NSF, the A. P. Sloan Foundation and a grant from the Texas Advanced Research Program.
Communicated by:
Ronald A. Fintushel
Article copyright:
© Copyright 2000
American Mathematical Society
