Vol3 and other exceptional hyperbolic 3-manifolds

Authors:
K. N. Jones and A. W. Reid

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2175-2185

MSC (2000):
Primary 57M50

Published electronically:
December 4, 2000

MathSciNet review:
1825931

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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****2.**A. Borel,*Commensurability classes and volumes of hyperbolic 3-manifolds*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**8**(1981), no. 1, 1–33. MR**616899****3.**Ted Chinburg and Eduardo Friedman,*The smallest arithmetic hyperbolic three-orbifold*, Invent. Math.**86**(1986), no. 3, 507–527. MR**860679**, 10.1007/BF01389265**4.**T. Chinburg, E. Friedman, K. N. Jones and A. W. Reid,*The arithmetic hyperbolic 3-manifold of smallest volume*, Ann. Scuola Norm. Sup. Pisa, to appear.**5.**David Gabai,*On the geometric and topological rigidity of hyperbolic 3-manifolds*, J. Amer. Math. Soc.**10**(1997), no. 1, 37–74. MR**1354958**, 10.1090/S0894-0347-97-00206-3**6.**D. Gabai, R. Meyerhoff and N. Thurston,*Homotopy hyperbolic 3-manifolds are hyperbolic*, Preprint.**7.**Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia,*A characterization of arithmetic subgroups of 𝑆𝐿(2,𝑅) and 𝑆𝐿(2,𝐶)*, Math. Nachr.**159**(1992), 245–270. MR**1237113**, 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**, 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****11.**Alan W. Reid,*A note on trace-fields of Kleinian groups*, Bull. London Math. Soc.**22**(1990), no. 4, 349–352. MR**1058310**, 10.1112/blms/22.4.349**12.**Alan W. Reid,*A non-Haken hyperbolic 3-manifold covered by a surface bundle*, Pacific J. Math.**167**(1995), no. 1, 163–182. MR**1318168****13.**Marie-France Vignéras,*Arithmétique des algèbres de quaternions*, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR**580949**

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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:
https://doi.org/10.1090/S0002-9939-00-05775-0

Keywords:
Hyperbolic 3-manifold,
arithmetic manifold,
homotopy hyperbolic 3-manifold

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