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Heights of simple loops and pseudo-Anosov homeomorphisms. (English) Zbl 0663.57010

Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 327-338 (1988).
[For the entire collection see Zbl 0651.00010.]
The height of a simple loop \(\ell\) in the boundary of a 3-dimensional handlebody H (of genus \(g>1)\) is the minimum value of -\(\chi\) (F) for an incompressible and non-boundary-parallel surface F properly imbedded in H-\(\ell\). Consider a set of 3g-3 proper discs in H whose boundaries give a pair-of-pants decomposition of \(\partial H\). The author proves that if h is a pseudo-Anosov diffeomorphism of \(\partial H\) whose stable lamination meets the set of discs in a sufficiently complicated way, then the heights \(h^ n(\ell)\) tend to \(\infty\) as n does. This implies that for any N, there is an m such that for all \(n>m\), the closed 3-manifold formed by a Heergaard diagram using \(h^ n\) contains no 2-sided incompressible surface of genus less than N.
The result about heights is proved using the space PL(\(\partial H)\) of projective measured laminations in \(\partial H\). By a geometric argument in the handlebody, the question is reduced to proving that the unstable lamination of h is not in the closure of any of the inductively-defined subspaces \(S_ n\) of PL(\(\partial H)\), where \(S_ 1\) is the set of simple loops disjoint from \(\ell\), and \(S_{k+1}\) is the set of simple loops each of which is disjoint from some element of \(S_ k\). Then, a pretty argument using the intersection number for measured laminations verifies this fact. The argument showing that the closed 3-manifolds do not contain incompressible surfaces of small genus makes use of some recent results of Casson and Gordon.
In a final section, the results are generalized to 3-manifolds with boundary.
Reviewer: D.McCullough

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57Q35 Embeddings and immersions in PL-topology
57R50 Differential topological aspects of diffeomorphisms

Citations:

Zbl 0651.00010