Commensurability of 1-cusped hyperbolic 3-manifolds

Authors:
Danny Calegari and Nathan M. Dunfield

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2955-2969

MSC (2000):
Primary 57M25, 57M50

DOI:
https://doi.org/10.1090/S0002-9947-02-02988-4

Published electronically:
February 25, 2002

MathSciNet review:
1895211

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.

**[AR]**I. R. Aitchison and J. H. Rubinstein,*Combinatorial cubings, cusps, and the dodecahedral knots*, Topology '90 (Columbus, OH, 1990), de Gruyter, Berlin, 1992, pp. 17-26. MR**93i:57016****[BNS]**Robert Bieri, Walter D. Neumann, and Ralph Strebel,*A geometric invariant of discrete groups*, Invent. Math.**90**(1987), no. 3, 451-477. MR**89b:20108****[Bor]**A. Borel,*Commensurability classes and volumes of hyperbolic -manifolds*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**8**(1981), no. 1, 1-33. MR**82j:22008****[Bro]**Kenneth S. Brown,*Trees, valuations, and the Bieri-Neumann-Strebel invariant*, Invent. Math.**90**(1987), no. 3, 479-504. MR**89e:20060****[BW]**Michel Boileau and Shicheng Wang,*Non-zero degree maps and surface bundles over*, J. Differential Geom.**43**(1996), no. 4, 789-806. MR**98g:57023****[BZa]**S. Boyer and X. Zhang,*On Culler-Shalen seminorms and Dehn filling*, Ann. of Math. (2)**148**(1998), no. 3, 737-801. MR**2000d:57028****[BZb]**Gerhard Burde and Heiner Zieschang,*Knots*, Walter de Gruyter & Co., Berlin, 1985. MR**87b:57004****[CGLS]**Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen,*Dehn surgery on knots*, Ann. of Math. (2)**125**(1987), no. 2, 237-300. MR**88a:57026****[CHW]**Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks,*A census of cusped hyperbolic -manifolds*, Math. Comp.**68**(1999), no. 225, 321-332. MR**99c:57035****[CS]**Marc Culler and Peter B. Shalen,*Varieties of group representations and splittings of -manifolds*, Ann. of Math. (2)**117**(1983), no. 1, 109-146. MR**84k:57005****[Dun]**Nathan M. Dunfield,*Alexander and Thurston norms of 3-manifolds fibering over the circle*, Pacific J. Math**200**(2001), no. 1, 43-58, arXiv:math.GT/9908050.**[G]**O. Goodman,*Snap*,`http://www.ms.unimelb.edu.au/ snap/`.**[HLM]**Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia,*On the arithmetic -bridge knots and link orbifolds and a new knot invariant*, J. Knot Theory Ramifications**4**(1995), no. 1, 81-114. MR**96a:57019****[HMW]**Craig D. Hodgson, G. Robert Meyerhoff, and Jeffrey R. Weeks,*Surgeries on the Whitehead link yield geometrically similar manifolds*, Topology '90 (Columbus, OH, 1990), de Gruyter, Berlin, 1992, pp. 195-206. MR**93i:57019****[HT]**A. Hatcher and W. Thurston,*Incompressible surfaces in -bridge knot complements*, Invent. Math.**79**(1985), no. 2, 225-246. MR**86g:57003****[Kal1]**E. Kaltofen,*Polynomial factorization*, Computer Algebra (B. Buchberger, G. Collins, and R. Loos, eds.), Springer Verlag, Heidelberg, 2 ed., 1982, pp. 95-113.**[Kal2]**Erich Kaltofen,*Polynomial factorization 1982-1986*, Computers in mathematics (Stanford, CA, 1986), Dekker, New York, 1990, pp. 285-309. MR**92f:12001****[Kir]**Rob Kirby,*Problems in low-dimensional topology*, Geometric topology (Athens, GA, 1993), Amer. Math. Soc., Providence, RI, 1997,`http://www.math.berkeley.edu/kirby/`, pp. 35-473. MR**98f:57001****[Lac]**Marc Lackenby,*Taut ideal triangulations of 3-manifolds*, Geom. Topol.**4**(2000), 369-395, arXiv:math.GT/0003132. MR**1 790 190****[LR]**D. D. Long and A. W. Reid,*Commensurability and the character variety*, Math. Res. Lett.**6**(1999), no. 5-6, 581-591. MR**2000m:57017****[Lub]**Alexander Lubotzky,*Subgroup growth and congruence subgroups*, Invent. Math.**119**(1995), no. 2, 267-295. MR**95m:20054****[McM]**Curtis T. McMullen,*The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology*, to appear in Ann. Sci. École. Norm. Sup. (4).**[Rei]**Alan W. Reid,*Arithmeticity of knot complements*, J. London Math. Soc. (2)**43**(1991), no. 1, 171-184. MR**92a:57011****[Ril]**Robert Riley,*Parabolic representations of knot groups. I*, Proc. London Math. Soc. (3)**24**(1972), 217-242. MR**45 #9313****[Rol]**Dale Rolfsen,*Knots and links*, Publish or Perish Inc., Berkeley, Calif., 1976, Mathematics Lecture Series, No. 7. MR**58 #24236****[Sha]**P. B. Shalen,*Representations of 3-manifold groups*, Handbook of geometric topology, Elsevier Press, to appear.**[Sta]**John Stallings,*On fibering certain -manifolds*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95-100. MR**28 #1600****[Thu]**William P. Thurston,*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357-381. MR**83h:57019****[Tur]**V. G. Turaev,*The Alexander polynomial of a three-dimensional manifold*, Math. USSR Sbornik**26**(1975), 313-329. MR**52 #4306****[W]**J. Weeks,*SnapPea*,`http://www.northnet.org/weeks/`.**[Wat]**Waterloo Maple Software,*Maple 6*, 2000.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M25,
57M50

Retrieve articles in all journals with MSC (2000): 57M25, 57M50

Additional Information

**Danny Calegari**

Affiliation:
Department of Mathematics, Harvard University, Cambridge Massachusetts 02138

Email:
dannyc@math.harvard.edu

**Nathan M. Dunfield**

Affiliation:
Department of Mathematics, Harvard University, Cambridge Massachusetts 02138

Email:
nathand@math.harvard.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02988-4

Keywords:
Virtual Fibration Conjecture,
commensurability,
Alexander polynomial,
character variety

Received by editor(s):
February 7, 2001

Received by editor(s) in revised form:
August 25, 2001

Published electronically:
February 25, 2002

Additional Notes:
Both authors were partially supported by the National Science Foundation.

Article copyright:
© Copyright 2002
American Mathematical Society