Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

<p style='text-indent:20px;'>We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Möbius band or a plane.</p><p style='text-indent:20px;'>Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-map (for <inline-formula><tex-math id="M2">\begin{document}$ m&gt;0 $\end{document}</tex-math></inline-formula> large) of a non-transitive Anosov flow <inline-formula><tex-math id="M3">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula> on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to <inline-formula><tex-math id="M4">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula>. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.</p>

Generating the twist subgroup by involutions

For a nonorientable surface, the twist subgroup is an index [Formula: see text] subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.