The kernel of the map from the handlebody group to Outer automorphisms of a free group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:36:41Z http://mathoverflow.net/feeds/question/22984 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22984/the-kernel-of-the-map-from-the-handlebody-group-to-outer-automorphisms-of-a-free The kernel of the map from the handlebody group to Outer automorphisms of a free group Jeffrey Giansiracusa 2010-04-29T14:35:10Z 2010-04-29T17:02:33Z <p>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 <em>handlebody group</em>. (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</p> <p><code>$H_g \to Out(F_g).$</code></p> <p>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?</p> <p>Does anyone know of a reference, proof, or counter example?</p> http://mathoverflow.net/questions/22984/the-kernel-of-the-map-from-the-handlebody-group-to-outer-automorphisms-of-a-free/22996#22996 Answer by Andy Putman for The kernel of the map from the handlebody group to Outer automorphisms of a free group Andy Putman 2010-04-29T15:52:42Z 2010-04-29T15:52:42Z <p>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.</p> http://mathoverflow.net/questions/22984/the-kernel-of-the-map-from-the-handlebody-group-to-outer-automorphisms-of-a-free/23006#23006 Answer by Sam Nead for The kernel of the map from the handlebody group to Outer automorphisms of a free group Sam Nead 2010-04-29T16:53:19Z 2010-04-29T17:02:33Z <p>This result is due to Luft; see "Actions of the homeotopy group of an orientable 3-dimensional handlebody".</p> <p>McCullough, in "Twist groups of compact 3-manifolds", proves that the twist group is not finitely generated and gives further references. </p> <p>EDIT. Ninja'ed (well, at least the first sentence).</p>