# The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the mapping class group of the genus $g$ surface $\partial K$.) The fundamental group of $K$ is a free group of rank $g$, so there is a homomorphism

$H_g \to Out(F_g).$

I've been thinking about this homomorphism and its kernel, and I've come to suspect that the kernel is generated by Dehn twists around curves in $\partial K$ that bound discs in $K$. These elements are all clearly contained in the kernel, but do they generate the entire kernel?

Does anyone know of a reference, proof, or counter example?

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May one ask why you are interested in this homomorphism? – Sam Nead Apr 29 '10 at 19:23
Well, I've been thinking about this map quite a lot recently in the context of modular operads. These sorts of groups make some very interesting operads. (For example, algebras over the operad of surface MCGs are infinite loop spaces by a theorem of Tillmann). These operads are related to topological conformal field theory and string topology and various other things. – Jeffrey Giansiracusa Apr 30 '10 at 6:45