Tightness and efficiency of irreducible automorphisms of handlebodies (Q2501677)

In order to attempt a classification of automorphisms of \(3\)-manifolds one is led to study automorphisms of handlebodies and compression bodies. Oertel showed that an automorphism \(f\) of a handlebody \(H\) is either periodic or reducible or irreducible. Here \(f\) is \textit{reducible} if there are (up to isotopy) an \(f\)-invariant essential compression body or essential annuli in \(H\) or \(H\) admits an \(f\)-invariant \(I\)-bundle structure. \(f\) is \textit{irreducible} if its restriction to \(\partial H\) is pseudo-Anosov and there is no closed surface \(F\) that is the interior boundary of an essential \(f\)-invariant compression body in \(H\). In the irreducible case Oertel showed that one can choose a concentric handlebody \(H_0\) in \({int}(H)\) such that \(f\) can be isotoped so that \(H_1=f(H_0)\) contains \(H_0\) in its interior, and \(\bigcup_{i\in\mathbb{Z}} f^i(H_0) = { int}(H)\). This leads to a two-dimensional measured lamination \((\Lambda , \mu)\) of \({ int}(H)\) and a transverse one-dimensional measured singular lamination \((\Omega,\nu)\) of \(H_0\) such that the leaves of \(\Lambda \) are open discs and fill \(H_0\), \(\Lambda \cup \partial H\) is closed in \(H\), \(\Lambda \cap \Omega\) is disjoint from the singular set of \(\Omega\), and after an isotopy of \(f\), \(f(\Lambda, \mu) =(\Lambda, \lambda \mu)\) and \(f(\Omega,\nu)=(\Omega, \lambda^{-1} \nu)\), for some \(\lambda>1\). The author addresses the problem of characterizing canonical laminations for a given irreducible \(f\). He proves that if \(\Lambda\) is tight then the growth rate \(\lambda\) is minimal . He shows that the converse is true under some technical conditions which hold for genus two handlebodies. As corollaries he obtains the results that if \(\Lambda\) is tight, then the growth rate \(\lambda\) is less than or equal to the growth rate of the restriction of \(f\) to \({\partial H}\) and the minimal growth \(\lambda_{\min}(f^n)\) is \((\lambda_{\min}(f))^n\), for any power \(f^n\).