An alternative approach to extending pseudo-Anosovs over compression bodies (Q498741)

The author gives an alternative proof of a theorem by \textit{I. Biringer} et al. [J. Topol. 6, No. 4, 1019--1042 (2013; Zbl 1293.57007)] concerning compressibility of pseudo-Anosov automorphisms. The proof shown in this paper is rather elementary, not using \(\delta\)-hyperbolic geometry, the curve complex, and Ahlfors-Bers theory of which the authors made use in the original proof. Let \(F\) denote a closed orientable surface of genus greater than one. Suppose a collection \(\mathcal{C}\) of disjoint, mutually non-isotopic, essential closed curves in \(F\) is given. The pair \((F, \mathcal{C})\) corresponds to a compression body \(M\) with the planar kernel \(\mathrm{Ker}(i_{\ast}:\pi_{1}(F)\rightarrow\pi_{1}(M))=\langle\mathcal{C}\rangle_{N}\), where \(i\) denotes the inclusion of \(F\) onto the exterior boundary of \(M\) and \(\langle\mathcal{C}\rangle_{N}\) the normal closure of the subgroup generated by \(\mathcal{C}\) in \(\pi_{1}(F)\). Assume the stable lamination of a pseudo-Anosov automorphism \(\varphi :F\rightarrow F\) is maximal, i.e., any complementary region is an ideal triangle, and bounds in \((F, \mathcal{C})\), i.e., it is the Hausdorff limit of curves bounding disks in the compression body \(M\). Also, \(M\) (or the pair \((F, \mathcal{C})\)) is assumed to be minimal, i.e., there is no inequivalent compression body \(N\subset M\) in which the lamination bounds. Then the theorem asserts that a power \(\varphi^{k}\) of \(\varphi\) extends over \(M\). The proof is based on the results in the papers by \textit{A. J. Casson} and \textit{D. D. Long} [Invent. Math. 81, 295--303 (1985; Zbl 0589.57008)] and by \textit{D. D. Long} [Duke Math. J. 56, No. 1, 1--16 (1988; Zbl 0655.57021)].