In a [paper of Casson and Gordon's](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=casson&s5=gordon&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3&mx-pid=722728) "A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983), no. 1, 119–137" they give a necessary criterion for a fibred knot to be a ribbon knot. 


The criterion is that the monodromy of the bundle has to extend to an automorphism of *some* handlebody (technicality, the fibre has a circle boundary so one caps-off the surface with a disc to get a closed surface). The handlebody is not *known* from simply the knot in advance, as it comes from a generalized Dehn lemma that they prove. 


My question:


* Q: Given an outer automorphism of the fundamental group of a closed orientable surface, is there an efficient procedure to determine if the automorphism extends to *some* handlebody that has the surface as boundary?


By the regular 3-manifold Dehn lemma, an automorphism of the surface extends to some handlebody if and only if it preserves the class of curves that bound discs in the handlebody -- which can be phrased entirely in terms of the homomorphism $\pi\_1 \Sigma \to \pi\_1 H$ where $\Sigma$ is the surface is the boundary of the handlebody $\Sigma = \partial H$. So this is easy to check. 


My question has the do with the case where you want to know if such a handlebody exists. Presumably there is a computable procedure for this, but it does not seem immediate to me. 



A procedure to determine if an automorphism of a closed 2-manifold extends to an automorphism of a handlebody
This problem is solved in the paper "Algorithmic compression of surface automorphisms" by Casson and Long. They remark that their motivation is (partly) the paper of Casson and Gordon. 


