Celebratio Mathematica

S. Boyer, C. M. Gordon, and X. Zhang: “Dehn fillings of large hyperbolic 3-manifolds,” J. Diff. Geom. 58 : 2 (2001), pp. 263–308. MR 1913944 Zbl 1042.57007 article

Let \( M \) be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus and which contains an essential closed surface \( S \). It is conjectured that 5 is an upper bound for the distance between two slopes on \( \partial M \) whose associated fillings are not hyperbolic manifolds. In this paper we verify the conjecture when the first Betti number of \( M \) is at least 2 by showing that given a pseudo-Anosov mapping class \( f \) of a surface and an essential simple closed curve \( \gamma \) in the surface, then 5 is an upper bound for the diameter of the set of integers \( n \) for which the composition of \( f \) with the \( n \)th power of a Dehn twist along \( \gamma \) is not pseudo-Anosov. This sharpens an inequality of Albert Fathi. For large manifolds \( M \) of first Betti number 1 we obtain partial results. Set \[ \mathcal{C}(S) = \bigl\{\text{slopes } r \mid \ker(\pi_1(S)\to \pi_1(M(r))) \neq \{1\} \bigr\}. \] A singular slope for \( S \) is a slope \( r_0\in \mathcal(S) \) such that any other slope in \( \mathcal{C}(S) \) is at most distance 1 from \( r_0 \). We prove that the distance between two exceptional filling slopes is at most 5 if either (i) there is a closed essential surface \( S \) in \( M \) with \( \mathcal{C}(S) \) finite, or (ii) there are singular slopes \( r_1\neq r_2 \) for closed essential surfaces \( S_1 \), \( S_2 \) in \( M \).

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S. Boyer, C. M. Gordon, and X. Zhang: “Reducible and finite Dehn fillings,” J. Lond. Math. Soc. (2) 79 : 1 (2009), pp. 72–84. MR 2472134 Zbl 1162.57015 ArXiv 0710.3786 article

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S. Boyer, C. M. Gordon, and X. Zhang: “Characteristic submanifold theory and toroidal Dehn filling,” Adv. Math. 230 : 4–6 (July–August 2012), pp. 1673–1737. MR 2927352 Zbl 1248.57004 ArXiv 1104.3321 article

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes \( \alpha \) and \( \beta \) on the boundary of a hyperbolic knot manifold \( M \) has been verified in all cases other than small Seifert filling slopes. In this paper, we verify it when \( \alpha \) is a small Seifert filling slope and \( \beta \) is a toroidal filling slope in the generic case where \( M \) admits no punctured-torus fiber or semi-fiber, and there is no incompressible torus in \( M(\beta) \) which intersects \( \partial M \) in one or two components. Under these hypotheses we show that \( \Delta(\alpha,\beta)\leq 5 \). Our proof is based on an analysis of the relationship between the topology of \( M \), the combinatorics of the intersection graph of an immersed disk or torus in \( M(\alpha) \) , and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in \( M \).

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S. Boyer, C. M. Gordon, and X. Zhang: “Dehn fillings of knot manifolds containing essential once-punctured tori,” Trans. Am. Math. Soc. 366 : 1 (2014), pp. 341–393. MR 3118399 Zbl 1290.57005 ArXiv 1109.5151 article

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let \( M \) be such a knot manifold and let \( \beta \) be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling \( M \) with slope \( \alpha \) produces a Seifert fibred manifold, then \( \Delta(\alpha,\beta )\leq 5 \). Furthermore we classify the triples \( (M;\alpha,\beta) \) when \( \Delta(\alpha,\beta)\geq 4 \). More precisely, when \( \Delta(\alpha,\beta)=5 \), then \( M \) is the (unique) manifold \( \textit{Wh}(-3/2) \) obtained by Dehn filling one boundary component of the Whitehead link exterior with slope \( -3/2 \), and \( (\alpha,\beta) \) is the pair of slopes \( (-5,0) \). Further, \( \Delta(\alpha,\beta)=4 \) if and only if \( (M;\alpha,\beta) \) is the triple \[ \Bigl(\textit{Wh}\bigl(\tfrac{-2n\pm 1}{n}\bigr);-4,0\Bigr) \] for some integer \( n \) with \( |n| > 1 \). Combining this with known results, we classify all hyperbolic knot manifolds \( M \) and pairs of slopes \( (\beta,\gamma) \) on \( \partial M \) where \( \beta \) is the boundary slope of an essential once-punctured torus in \( M \) and \( \gamma \) is an exceptional filling slope of distance 4 or more from \( \beta \). Refined results in the special case of hyperbolic genus one knot exteriors in \( S^3 \) are also given.

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Xingru Zhang	Related

S. Boyer, C. M. Gordon, and X. Zhang: Dehn fillings of knot manifolds containing essential twice-punctured tori. Mem. Amer. Math. Soc. 295. 2024. ArXiv 2004.04219 book

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Cameron McAllan Gordon

Working with Cameron: from exceptional surgeries to the L-space conjecture

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