Hyperbolic 3-manifolds and surface automorphisms
The authors exploit the close interplay between hyperbolic 3-manifolds and pseudo-Anosov maps which arises via the mapping torus construction. Namely, let F be a surface, \(\theta\) its automorphism and \(M(\theta)=F[0,1]/\sim\) where (x,0)\(\sim (\theta (x),1)\) the mapping torus with the monodromy \(\theta\). If \(T: F\to F\) is a Dehn twist about a simple closed curve, c, on F, then \(M(T^ r\theta)\) is obtained from M(\(\theta)\) by doing a (1,r) Dehn surgery on c. Now assuming that M(\(\theta)\)-c is hyperbolic one gets, using a result of \textit{W. P. Thurston} [''The geometry and topology of 3-manifolds'', preprint (1979)], that all but finitely many Dehn surgeries on c produce hyperbolic manifolds. Using this the authors prove: Theorem 1. Let \(\theta\) : \(F\to F\) be a pseudo-Anosov map and \(T: F\to F\) be a Dehn twist about an essential simple closed curve, c, in F. Then (i) for all but finitely many r, \(T^ r\theta\) is isotopic to a pseudo-Anosov map; (ii) there is a K, such that for all \(r\geq K\), the maps \(T^ r\theta\) are all nonconjugate. Furthermore the authors partially determine the behavior of the unstable lamination of \(T^ r\theta\) and approximate the dilatation of \(T^ r\theta\).
scientific article; zbMATH DE number 3982076