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

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

2010-04-29T14:35:10Z

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?

Answer by Andy Putman for The kernel of the map from the handlebody group to Outer automorphisms of a free group

2010-04-29T15:52:42Z

The answer is yes. I'm not sure who this is due to, but one nice reference is section 3.9 of Fomenko-Matveev's book "Algorithmic and Computer Methods for 3-Manifolds". It's not stated in precisely the form you stated it, but it can be easily extracted from there.

Answer by Sam Nead for The kernel of the map from the handlebody group to Outer automorphisms of a free group

2010-04-29T16:53:19Z

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).