Characteristic subsurfaces, character varieties and Dehn fillings. (English) Zbl 1147.57002
For a one-cusped hyperbolic \(3\)-manifold \(M\), there are only finitely many slopes on the boundary torus of \(M\) which yield non-hyperbolic \(3\)-manifolds by Dehn filling. The results in this paper give new upper bounds for the distance \(\Delta(\alpha,\beta)\) between two such exceptional slopes \(\alpha\) and \(\beta\) in several special cases.
As usual, let \(M(\gamma)\) denote the resulting closed manifold obtained by \(\gamma\)-Dehn filling on \(M\). The first result claims that if \(M(\beta)\) is reducible and \(M(\alpha)\) has finite fundamental group, then \(\Delta(\alpha,\beta)\leq 2\). Moreover, some constraints are given when \(\Delta=2\). This is an improvement of the previouly known bound by S. Boyer and X. Zhang [Ann. Math. (2) 148, No. 3, 737–801 (1998; Zbl 1007.57016)]. (Finally, the case where \(\Delta=2\) is eliminated by S. Boyer, C. Gordon and X. Zhang [Reducible and finite Dehn fillings, preprint, 2007].)
Suppose that \(M(\beta)\) is reducible and \(\beta\) is a strict boundary slope. If \(M(\alpha)\) is very small, then \(\Delta(\alpha,\beta)\leq 3\). A closed \(3\)-manifold is said to be very small if its fundamental group has no non-abelian free subgroup.
If \(M(\alpha)\) admits a \(\pi_1\)-injective immersed torus, then \(\Delta(\alpha,\beta)\leq 4\).
In general, C. McA. Gordon [Jones, Vaughan F. R. (ed.) et al., Knot theory. Proceedings of the mini-semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42, 129–144 (1998; Zbl 0916.57016)] conjectures that \(\Delta(\alpha,\beta)\leq 8\) for any two exceptional slopes, and there are only four specific manifolds which have a pair of exceptional slopes with distance greater than \(5\). The results in the paper under review can imply the restricted solution of Gordon’s conjecture.
Most of the arguments are based on \(PSL_2(\mathbb{C})\)-character variety theory developed in Marc Culler and Peter B. Shalen [Ann. Math. (2) 117, 109–146 (1983; Zbl 0529.57005)], S. Boyer and X. Zhang [Ann. Math. (2) 148, No. 3, 737–801 (1998; Zbl 1007.57016), and J. Differ. Geom. 59, No. 1, 87–176 (2001; Zbl 1030.57024)], etc.
As usual, let \(M(\gamma)\) denote the resulting closed manifold obtained by \(\gamma\)-Dehn filling on \(M\). The first result claims that if \(M(\beta)\) is reducible and \(M(\alpha)\) has finite fundamental group, then \(\Delta(\alpha,\beta)\leq 2\). Moreover, some constraints are given when \(\Delta=2\). This is an improvement of the previouly known bound by S. Boyer and X. Zhang [Ann. Math. (2) 148, No. 3, 737–801 (1998; Zbl 1007.57016)]. (Finally, the case where \(\Delta=2\) is eliminated by S. Boyer, C. Gordon and X. Zhang [Reducible and finite Dehn fillings, preprint, 2007].)
Suppose that \(M(\beta)\) is reducible and \(\beta\) is a strict boundary slope. If \(M(\alpha)\) is very small, then \(\Delta(\alpha,\beta)\leq 3\). A closed \(3\)-manifold is said to be very small if its fundamental group has no non-abelian free subgroup.
If \(M(\alpha)\) admits a \(\pi_1\)-injective immersed torus, then \(\Delta(\alpha,\beta)\leq 4\).
In general, C. McA. Gordon [Jones, Vaughan F. R. (ed.) et al., Knot theory. Proceedings of the mini-semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42, 129–144 (1998; Zbl 0916.57016)] conjectures that \(\Delta(\alpha,\beta)\leq 8\) for any two exceptional slopes, and there are only four specific manifolds which have a pair of exceptional slopes with distance greater than \(5\). The results in the paper under review can imply the restricted solution of Gordon’s conjecture.
Most of the arguments are based on \(PSL_2(\mathbb{C})\)-character variety theory developed in Marc Culler and Peter B. Shalen [Ann. Math. (2) 117, 109–146 (1983; Zbl 0529.57005)], S. Boyer and X. Zhang [Ann. Math. (2) 148, No. 3, 737–801 (1998; Zbl 1007.57016), and J. Differ. Geom. 59, No. 1, 87–176 (2001; Zbl 1030.57024)], etc.
Reviewer: Masakazu Teragaito (Hiroshima)
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 57M99 | General low-dimensional topology |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
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