## Presentation length and Simon’s conjecture

HTML articles powered by AMS MathViewer

- by Ian Agol and Yi Liu PDF
- J. Amer. Math. Soc.
**25**(2012), 151-187 Request permission

## Abstract:

In this paper, we show that any knot group maps onto at most finitely many knot groups. This gives an affirmative answer to a conjecture of J. Simon. We also bound the diameter of a closed hyperbolic 3-manifold linearly in terms of the presentation length of its fundamental group, improving a result of White.## References

- Michel Boileau, Steve Boyer, Alan W. Reid, and Shicheng Wang,
*Simon’s conjecture for two-bridge knots*, Comm. Anal. Geom.**18**(2010), no. 1, 121–143. MR**2660460**, DOI 10.4310/CAG.2010.v18.n1.a5 - Michel Boileau, Hyam Rubinstein, and Shicheng Wang,
*Finiteness of $3$-manifolds associated with non-zero degree mappings*, preprint, 2005, math.GT/0511541 . - Michel Boileau and Richard Weidmann,
*The structure of 3-manifolds with two-generated fundamental group*, Topology**44**(2005), no. 2, 283–320. MR**2114709**, DOI 10.1016/j.top.2004.10.008 - Ryan Budney,
*JSJ-decompositions of knot and link complements in $S^3$*, Enseign. Math. (2)**52**(2006), no. 3-4, 319–359. MR**2300613** - C. Cao, F. W. Gehring, and G. J. Martin,
*Lattice constants and a lemma of Zagier*, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 107–120. MR**1476983**, DOI 10.1090/conm/211/02816 - Daniel E. Cohen,
*Combinatorial group theory. A topological approach*, Queen Mary College Mathematics Notes, Queen Mary College, Department of Pure Mathematics, London, 1978. MR**506780** - Daryl Cooper,
*The volume of a closed hyperbolic $3$-manifold is bounded by $\pi$ times the length of any presentation of its fundamental group*, Proc. Amer. Math. Soc.**127**(1999), no. 3, 941–942. MR**1646313**, DOI 10.1090/S0002-9939-99-05190-4 - 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**881270**, DOI 10.2307/1971311 - Keiichi Horie, Teruaki Kitano, Mineko Matsumoto, and Masaaki Suzuki,
*A partial order on the set of prime knots with up to $11$ crossings*, arXiv:0906.3943 . - Jim Hoste and Patrick D. Shanahan,
*Epimorphisms and boundary slopes of 2-bridge knots*, Algebr. Geom. Topol.**10**(2010), no. 2, 1221–1244. MR**2653061**, DOI 10.2140/agt.2010.10.1221 - Rob Kirby (ed.),
*Problems in low-dimensional topology*, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473. MR**1470751**, DOI 10.1090/amsip/002.2/02 - Teruaki Kitano and Masaaki Suzuki,
*Twisted Alexander polynomials and a partial order on the set of prime knots*, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 307–321. MR**2508212**, DOI 10.2140/gtm.2008.13.307 - Donghi Lee and Makoto Sakuma,
*Epimorphisms between $2$-bridge link groups: Homotopically trivial simple loops on $2$-bridge spheres*, arXiv:1004.2571 . - Chris Leininger,
*Simon’s conjecture for fibered knots*, arXiv:0907.3263v1 . - Gaven J. Martin,
*Triangle subgroups of Kleinian groups*, Comment. Math. Helv.**71**(1996), no. 3, 339–361. MR**1418942**, DOI 10.1007/BF02566424 - Darryl McCullough,
*Compact submanifolds of $3$-manifolds with boundary*, Quart. J. Math. Oxford Ser. (2)**37**(1986), no. 147, 299–307. MR**854628**, DOI 10.1093/qmath/37.3.299 - Tomotada Ohtsuki, Robert Riley, and Makoto Sakuma,
*Epimorphisms between 2-bridge link groups*, The Zieschang Gedenkschrift, Geom. Topol. Monogr., vol. 14, Geom. Topol. Publ., Coventry, 2008, pp. 417–450. MR**2484712**, DOI 10.2140/gtm.2008.14.417 - C. D. Papakyriakopoulos,
*On Dehn’s lemma and the asphericity of knots*, Ann. of Math. (2)**66**(1957), 1–26. MR**90053**, DOI 10.2307/1970113 - Shawn Rafalski,
*Immersed turnovers in hyperbolic 3-orbifolds*, Groups Geom. Dyn.**4**(2010), no. 2, 333–376. MR**2595095**, DOI 10.4171/GGD/86 - G. P. Scott,
*Compact submanifolds of $3$-manifolds*, J. London Math. Soc. (2)**7**(1973), 246–250. MR**326737**, DOI 10.1112/jlms/s2-7.2.246 - Zlil Sela,
*Diophantine geometry over groups. I. Makanin-Razborov diagrams*, Publ. Math. Inst. Hautes Études Sci.**93**(2001), 31–105. MR**1863735**, DOI 10.1007/s10240-001-8188-y - Daniel S. Silver and Wilbur Whitten,
*Knot group epimorphisms, II*, 2008, arXiv:0806.3223 . - Daniel S. Silver and Wilbur Whitten,
*Hyperbolic covering knots*, Algebr. Geom. Topol.**5**(2005), 1451–1469. MR**2186104**, DOI 10.2140/agt.2005.5.1451 - Daniel S. Silver and Wilbur Whitten,
*Knot group epimorphisms*, J. Knot Theory Ramifications**15**(2006), no. 2, 153–166. MR**2207903**, DOI 10.1142/S0218216506004373 - Teruhiko Soma,
*The Gromov invariant of links*, Invent. Math.**64**(1981), no. 3, 445–454. MR**632984**, DOI 10.1007/BF01389276 - William P. Thurston,
*The geometry and topology of $3$-manifolds*, Lecture notes from Princeton University, 1978–80, http://library.msri.org/books/gt3m/ . - Richard Weidmann,
*The Nielsen method for groups acting on trees*, Proc. London Math. Soc. (3)**85**(2002), no. 1, 93–118. MR**1901370**, DOI 10.1112/S0024611502013473 - Matthew E. White,
*A diameter bound for closed, hyperbolic $3$-manifolds*, 2001, math.GT/0104192 .

## Additional Information

**Ian Agol**- Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840
- MR Author ID: 671767
- ORCID: 0000-0002-4254-8483
- Email: ianagol@math.berkeley.edu
**Yi Liu**- Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840
- MR Author ID: 945775
- Email: yliu@math.berkeley.edu
- Received by editor(s): July 12, 2010
- Received by editor(s) in revised form: April 22, 2011
- Published electronically: July 12, 2011
- Additional Notes: The first and second authors were partially supported by NSF grant DMS-0806027
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**25**(2012), 151-187 - MSC (2010): Primary 57Mxx
- DOI: https://doi.org/10.1090/S0894-0347-2011-00711-X
- MathSciNet review: 2833481