The Big Dehn Surgery Graph and the link of \(S^3\) (Q2795853)

In the paper under review, the authors study an unoriented graph \(\mathcal{B}\) introduced by W. Thurston in 2004, and ask some interesting questions about it. This graph is constructed as follows: a vertex \(v_{M}\) for each closed, orientable \(3\)-manifold \(M\), and an edge between two different vertices \(v_M\) and \(v_{M'}\) if there exists a knot \(K\subset M\) such that \(M'\) is obtained by non-trivial Dehn surgery along \(K\) in \(M\). If there is another knot \(K_1\) in \(M\) that produces \(M'\), only one edge between \(v_M\) and \(v_{M'}\) is considered.NEWLINENEWLINEBy using the proof given by \textit{W. B. R. Lickorish} [Ann. Math. (2) 76, 531--540 (1962; Zbl 0106.37102)], the authors endow \(\mathcal{B}\) with a metric. A shortest path from \(v_{S^3}\) to \(v_M\) in \(\mathcal{B}\) counts the minimum number of components needed for a link in \(S^3\) to admit \(M\) as a surgery. They call this the Lickorish path length (known as surgery distance) and denote this function by \(p_L(v_{M_1}, v_{M_2})\), i.e., the number of edges in a shortest edge path between \(v_{M_1}\) and \(v_{M_2}\). It is proved that this graph is connected, has infinite valence and infinite diameter. Some subgraphs of \(\mathcal{B}\) are considered. Let \(lk (v) = \langle w : p_L (v,w) = 1 \rangle\) be the link of \(v\), i.e., the subgraph of \(\mathcal{B}\) generated by the vertices \(w\) which are at Lickorish path length \(1\) from \(v\). It is proved that the link of \(S^3\) in \(\mathcal{B}\) is connected and of bounded diameter.NEWLINENEWLINESome questions arise: is the link of any vertex in \(\mathcal{B}\) connected? of bounded diameter? The authors note that a negative answer would lead to an obstruction to automorphisms of the graph that do not fix \(S^3\). It is asked: does the graph \(\mathcal{B}\) admit a non-trivial automorphism? Further it is stated that an answer to this question would lead to a better understanding of how the Dehn surgery structure of a manifold relates to the homeomorphism type. Also, the following problem is posed: Characterize the vertices in the link of \(S^3\). An infinite family of hyperbolic manifolds is given none of which can be realized by surgery on knots in \(S^3\), and such that these manifolds have weight one fundamental groups. Previous examples were given by \textit{S. Boyer} and \textit{D. Lines} [J. Reine Angew. Math. 405, 181--220 (1990; Zbl 0691.57004)], who exhibited a set of small Seifert fibered spaces which are of weight one but can not be obtained by surgery along a knot in \(S^3\). Associated to any knot in \(M\) is a \(\mathbb{K}_{\infty}\), the complete graph on infinitely many vertices. If \(K\) is a knot in a \(3\)-manifold \(M\), denote by \((M)_{\infty}^{K} = \langle \{ v_{M(K;r)}\}\rangle\) where \(\{M(K;r)\}\) is the set of \(3\)-manifolds obtained from \(M\) via Dehn surgery on \(K\). It is proved that \((M)_{\infty}^K\) is a \(\mathbb{K}_{\infty}\), and that if \(M\backslash K\) and \(M' \backslash K'\) are hyperbolic and not homeomorphic, then \((M)_{\infty}^K\) and \((M')_{\infty}^{K'}\) have at most finitely many vertices in common. Furthermore, there is a \(\mathbb{K}_{\infty}\) of small Seifert-fibered spaces that does not come from surgery along a one cusped manifold.NEWLINENEWLINEIt is proved that: 1) If \(M\) is a \(Solv\) torus bundle, then \(p_L(M, S^3) \leq5\) ; 2) If \(M\) is the union of twisted \(I\) bundles over the Klein bottle admitting an orientable \(Solv\) structure, then \(p_L(M, S^3)\leq 3\). It follows that if \(M\) admits a \(Nil\), \(\mathbb{E}^3\), \(S^2\times \mathbb{R}\), \(S^3\) or \(Solv\) geometry, then \(p_L(M,S^3)\leq 5\). It is shown that if \(M_1\) and \(M_2\) are closed orientable \(3\)-manifolds and \(0\leq m\leq n\) and \(p\) a prime, and if there are epimorphisms from \(\pi _1 (M_1) \) to \(({\mathbb{Z}} / p{\mathbb{Z}})^n\), and from \(\pi _1 (M_2) \) to \(({\mathbb{Z}} / p{\mathbb{Z}})^m\) respectively, but there is not an epimorphism from \(\pi _1 (M_2)\) to \(({\mathbb{Z}} / p{\mathbb{Z}})^{m+1}\), then \(p_L(M_1, M_2) \geq n-m\).NEWLINENEWLINEAnother interesting subgraph studied in the paper under review is the subgraph of hyperbolic manifolds. Let \(\mathcal {B}_H \) be the subgraph of \(\mathcal{B}\) such that the vertices correspond to closed hyperbolic \(3\)-manifolds, and there is an edge between two vertices \(v_M\) and \(v_N\) exactly when there is a one-cusped hyperbolic \(3\)-manifold \(P\) with two fillings homeomorphic to \(M\) and \(N\). It is conjectured that the combinatorics of this subgraph may reveal more of geometry and topology than the full graph. It is shown that \(\mathcal{B}_H\) has infinite valence, has infinite diameter and is connected, and that if \(M_0\) and \(M_n\) are closed hyperbolic \(3\)-manifolds such that the associated vertices \(v_{M_0}\) and \(v_{M_n}\) are connected via a path of length \(n\) in \(\mathcal {B}\) then \(v_{M_0}\) and \(v_{M_n}\) are connected via a path of length \(n+2\) in \(\mathcal{B}_H\). A geodesic metric space is \(\delta\)-hyperbolic if every side of a geodesic triangle is contained in a \(\delta\)-neighborhood of the union of the other two sides. A \(k\)-quasi-flat in a metric space \(X\) is a subset of \(X\) that is quasi-isometric to \(\mathbb{E}^k\). The authors construct quasi-flats in \(\mathcal{B}\) and \(\mathcal{B}_H\), showing that these spaces are not \(\delta\)-hyperbolic.