A complex of incompressible surfaces for handlebodies and the mapping class group (Q714975)

Let \(H_g\) be a compact orientable handlebody of genus \(g\), and let \(F=\partial H_g\). We can associate to \(H_g\) several complexes. First, \({\mathcal C} (F)\) denotes the complex of curves of \(F\), introduced by \textit{W. J. Harvey} [Ann. Math. Stud. 97, 245--251 (1981; Zbl 0461.30036)], which has as vertices the isotopy classes of essential simple closed curves in \(F\), and a collection of vertices spans a simplex when any two of the vertices can be represented by disjoint curves. By \({\mathcal D} (H_g)\) denote the disk complex of \(H_g\), defined by \textit{D. McCullough} [J. Differ. Geom. 33, No. 1, 1--65 (1991; Zbl 0721.57008)], which has as vertices the isotopy classes of compressing disks for \(H_g\), and a collection of vertices spans a simplex when any two of the vertices can be represented by disjoint disks. In the paper under review a third complex is defined. Let \({\mathcal I} (H_g)\) be the complex whose vertices are the isotopy classes of compressing disks for \(\partial H_g\) and the isotopy classes of properly embedded boundary-parallel incompressible annuli and pair of pants in \(H_g\), and again a collection of vertices spans a simplex when any two of the vertices can be represented by disjoint surfaces. The complex \({\mathcal C} (F)\) can be considered as a subcomplex of \({\mathcal I} (H_g)\), where a vertex of \({\mathcal C} (F)\) is identified with a compressing disk, if the corresponding curve bounds a disk in \(H_g\), or with an essential annulus otherwise. There is a natural homomorphism \(A\) from the mapping class group of \(H_g\) to the group of automorphisms of \({\mathcal I} (H_g)\). The main result of the paper shows that for \(g\geq 2\), the map \(A\) is onto, i.e., any automorphism of \({\mathcal I} (H_g)\) is induced by a homeomorphism of \(H_g\), for \(g\geq 3\), the map is injective, and for \(g=2\), \(A\) has a \(\mathbb{Z}_2\)-kernel generated by the hyper-elliptic involution. It is also shown that the complex \({\mathcal I} (H_g)\) is \(\delta\)-hyperbolic in the sense of Gromov.