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?



The kernel of the map from the handlebody group to Outer automorphisms of a free group
This result is due to Luft; see "Actions of the homeotopy group of an orientable 3-dimensional handlebody".


McCullough, in "Twist groups of compact 3-manifolds", proves that the twist group is not finitely generated and gives further references. 


EDIT. Ninja'ed (well, at least the first sentence).


