TITLE: extension of surface homeomorphism
QUESTION [15 upvotes]: Can anyone give me a reference (or proof sketch) for the fact that there are psuedo-Anosov diffeomorphisms of closed hyperbolic surfaces which do not extend over any handlebody? Thanks.

REPLY [2 votes]: I accidentally found this Rice University thesis (never published, it seems) which studies the question in some depth:
Author Jamie Bradley Jorgensen
Title Surface homeomorphisms that do not extend to any handlebody and the Johnson Filtration.