Exceptional Dehn filling

This Research in Team workshop focused on several problems in the theory of exceptional Dehn fillings in 3dimensional topology. Dehn filling is the construction in which you take a 3-manifoldM , with a distinguished torus boundary component T , and glue a solid torus V to M via some homeomorphism from ∂V to T . The resulting manifold depends only on the isotopy class (slope) α on T that is identified with the boundary of a meridian disk of V , so we denote it by M(α). The construction goes back to Dehn in 1910, who introduced it in the special case where M is the exterior of a knot in S. The Lickorish-Wallace theorem of the early 1960’s showed that any closed, connected, orientable 3-manifold can be obtained by Dehn filling the boundary tori of the exterior of some link in S. Consequently, many of the basic problems in 3-manifold topology can been analysed in terms of the operation. Renewed interest in the construction arose with the ground-breaking work of Thurston in the 1970’s, who used it to study hyperbolic geometric structures on 3-manifolds. In particular, Thurston showed that if M is hyperbolic then M(α) is also hyperbolic for all but finitely many slopes α on T . When M is hyperbolic but M(α) is not, one says that (M ;α) is exceptional. Although it is clear that one cannot hope to classify all exceptional (M ;α)’s, it turns out that it is relatively rare for a hyperbolic 3-manifold to have two distinct exceptional slopes α and β on T , and it is not too unreasonable to try to classify all such (M ;α, β)’s. This has usually been approached by considering the various different ways in which M(α) and M(β) can fail to be hyperbolic, and there has been a lot of progress along these lines. The cases about which least is known is when the boundary of M is a torus and one of the fillings, say M(β), is a small Seifert fiber space. This was the focus of the workshop