Rank and genus of 3-manifolds

Li, Tao

doi:10.1090/S0894-0347-2013-00767-5

Rank and genus of 3-manifolds
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by Tao Li;

J. Amer. Math. Soc. 26 (2013), 777-829

DOI: https://doi.org/10.1090/S0894-0347-2013-00767-5

Published electronically: February 27, 2013

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Abstract:

We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.

References

Miklós Abért and Nikolay Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem. arXiv:math/0701361
David Bachman, Saul Schleimer, and Eric Sedgwick, Sweepouts of amalgamated 3-manifolds, Algebr. Geom. Topol. 6 (2006), 171–194. MR 2199458, DOI 10.2140/agt.2006.6.171
M. Boileau and H. Zieschang, Heegaard genus of closed orientable Seifert $3$-manifolds, Invent. Math. 76 (1984), no. 3, 455–468. MR 746538, DOI 10.1007/BF01388469
A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275–283. MR 918537, DOI 10.1016/0166-8641(87)90092-7
I. A. Grushko, On the bases of a free product of groups. Matematicheskii Sbornik 8 (1940), 169–182.
Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Studies in Mathematics, Vol. 5, Math. Assoc. America, 1968, pp. 39–98. MR 224071
Wolfgang Haken, Various aspects of the three-dimensional Poincaré problem, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham Publishing Co., Chicago, IL, 1970, pp. 140–152. MR 273624
A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373–377. MR 658066
Allen Hatcher, Basic Topology of Three Manifolds. Online notes, available at http://www.math.cornell.edu/hatcher/.
A. Hatcher and W. Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246. MR 778125, DOI 10.1007/BF01388971
John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657. MR 1838999, DOI 10.1016/S0040-9383(00)00033-1
Tsuyoshi Kobayashi, Classification of unknotting tunnels for two bridge knots, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 259–290. MR 1734412, DOI 10.2140/gtm.1999.2.259
Marc Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109 (2004), 139–145. MR 2113191, DOI 10.1007/s10711-004-6553-y
Marc Lackenby, The asymptotic behaviour of Heegaard genus, Math. Res. Lett. 11 (2004), no. 2-3, 139–149. MR 2067463, DOI 10.4310/MRL.2004.v11.n2.a1
Tao Li, Heegaard surfaces and measured laminations. I. The Waldhausen conjecture, Invent. Math. 167 (2007), no. 1, 135–177. MR 2264807, DOI 10.1007/s00222-006-0009-y
Tao Li, Heegaard surfaces and measured laminations. II. Non-Haken 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 3, 625–657. MR 2220101, DOI 10.1090/S0894-0347-06-00520-0
Tao Li, Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 1119–1134. MR 2336252, DOI 10.2140/agt.2007.7.1119
Tao Li, On the Heegaard splittings of amalgamated 3-manifolds, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 157–190. MR 2408246, DOI 10.2140/gtm.2007.12.157
Tao Li, Heegaard surfaces and the distance of amalgamation, Geom. Topol. 14 (2010), no. 4, 1871–1919. MR 2680206, DOI 10.2140/gt.2010.14.1871
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3
Yoav Moriah and Hyam Rubinstein, Heegaard structures of negatively curved $3$-manifolds, Comm. Anal. Geom. 5 (1997), no. 3, 375–412. MR 1487722, DOI 10.4310/CAG.1997.v5.n3.a1
Hossein Namazi and Juan Souto, Heegaard splittings and pseudo-Anosov maps, Geom. Funct. Anal. 19 (2009), no. 4, 1195–1228. MR 2570321, DOI 10.1007/s00039-009-0025-3
Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
Grisha Perelman, Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109
Grisha Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245
Yo’av Rieck, Heegaard structures of manifolds in the Dehn filling space, Topology 39 (2000), no. 3, 619–641. MR 1746912, DOI 10.1016/S0040-9383(99)00026-9
Yo’av Rieck and Eric Sedgwick, Persistence of Heegaard structures under Dehn filling, Topology Appl. 109 (2001), no. 1, 41–53. MR 1804562, DOI 10.1016/S0166-8641(99)00147-9
Martin Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135–147. MR 1648310, DOI 10.1016/S0166-8641(97)00184-3
Martin Scharlemann, Proximity in the curve complex: boundary reduction and bicompressible surfaces, Pacific J. Math. 228 (2006), no. 2, 325–348. MR 2274524, DOI 10.2140/pjm.2006.228.325
Martin Scharlemann, Heegaard splittings of compact 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 921–953. MR 1886684
Martin Scharlemann and Jennifer Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999), no. 2, 265–270. MR 1660345, DOI 10.1016/S0040-9383(98)00002-0
Martin Scharlemann and Maggy Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593–617. MR 2224466, DOI 10.2140/gt.2006.10.593
Martin Scharlemann and Abigail Thompson, Thin position for $3$-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231–238. MR 1282766, DOI 10.1090/conm/164/01596
Jennifer Schultens, The classification of Heegaard splittings for (compact orientable surface)$\,\times \, S^1$, Proc. London Math. Soc. (3) 67 (1993), no. 2, 425–448. MR 1226608, DOI 10.1112/plms/s3-67.2.425
Jennifer Schultens and Richard Weidman, On the geometric and the algebraic rank of graph manifolds, Pacific J. Math. 231 (2007), no. 2, 481–510. MR 2346507, DOI 10.2140/pjm.2007.231.481
Peter B. Shalen, Hyperbolic volume, Heegaard genus and ranks of groups, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 335–349. MR 2408254, DOI 10.2140/gtm.2007.12.335
Juan Souto, The rank of the fundamental group of certain hyperbolic 3-manifolds fibering over the circle, The Zieschang Gedenkschrift, Geom. Topol. Monogr., vol. 14, Geom. Topol. Publ., Coventry, 2008, pp. 505–518. MR 2484715, DOI 10.2140/gtm.2008.14.505
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
Friedhelm Waldhausen, Some problems on $3$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, RI, 1978, pp. 313–322. MR 520549
J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. (2) 37 (1936), no. 4, 782–800. MR 1503309, DOI 10.2307/1968618