×

The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of minimal pseudo-Anosov dilatations. (English) Zbl 1365.57019

Let \(\Sigma_{g,n}\) be the orientable surface of genus \(g\) with \(n\) punctures, and \(\delta_{g,n}\) the minimal dilatation of a pseudo-Anosov element of the mapping class group of \(\Sigma_{g,n}\). The authors study the asymptotic behaviour of the logarithm of \(\delta_{g,n}\) for fixed \(g\geq 2\) and varying \(n\). The starting point is the result of C. Y. Tsai [Geom. Topol. 13, No. 4, 2253–2278 (2009; Zbl 1204.37043)] who showed that for any fixed \(g\geq 2\), \(\log\delta_{g,n}\) has the growth order of \(\frac{\log n}{n}\). The main result of the paper under review asserts that given \(g\geq 2\) there exists a sequence \(\{n_i\}_{i=1}^\infty\) with \(n_i\to\infty\) such that \[ \limsup_{i\to\infty}\frac{n_i\log\delta_{g,n_i}}{\log{n_i}}\leq 2. \] If \(g\) satisfies an additional assumption, namely \(2g+1\) is relatively prime to \(s\) or \(s+1\) for each \(0\leq s\leq g\), then one can take the sequence \(\{n_i\}_{i=1}^\infty\) above to be the sequence of natural numbers \(\{n\}_{n=1}^\infty\)
The proof is based on the theory of fibered faces of hyperbolic 3-manifolds, introduced in [W. Thurston, Mem. Am. Math. Soc. 339, 99–130 (1986; Zbl 0585.57006)], applied to one particular 3-manifold \(N\), called the “magic manifold”, defined as the complement of a certain chain-link with \(3\) components. The authors find a suitable family of fibrations over the circle of \(N\) (or a manifold obtained from \(N\) by Dehn filling two cusps) whose fibers are surfaces of genus \(g\) with varying number of punctures. The monodromies of these fibrations are pseudo-Anosov, and their dilatations are roots of certain polynomials computed in [E. Kin and M. Takasawa, Commun. Anal. Geom. 19, No. 4, 705–758 (2011; Zbl 1251.37047)].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B40 Topological entropy
PDFBibTeX XMLCite
Full Text: arXiv Euclid