Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles
For an orientable closed surface \(F\) of genus \(g\geq 1\) and an automorphism \(f:F\to F\), denote by \(M(F,f)\) the surface bundle with fiber \(F\) and monodromy \(f\). Work of Thurston implies that for \(g\geq 2\) the surface bundle \(M(F,f)\) is a hyperbolic \(3\)-manifold if and only if \(f\) is a pseudo-Anosov map (in the paper under review the definition of a pseudo-Anosov map is slightly different from the standard one: it refers to a map that is isotopic to a pseudo-Anosov map in the standard sense). Let now \(F'\) and \(F\) be two closed orientable surfaces of genus \(g'>g\geq 1\), respectively, and let \(f:F\to F\) be a pseudo-Anosov map. The main result states that \(f\) can be ``extended'' to a pseudo-Anosov mapping \(f':F'\to F'\) so that there is a fiber-preserving degree one map \(M(F',f')\to M(F,f)\) between the hyperbolic surface bundles. Furthermore, the ``extension'' can be chosen in such a way that the first Betti numbers of the two surface bundles agree.
scientific article; zbMATH DE number 5162377