Finiteness Properties for a Subgroup of the Pure Symmetric Automorphism Group

Let Fn be the free group on n generators, and let PΣn be the group of automorphisms of Fn that send each generator to a conjugate of itself. The kernel Kn of the homomorphism PΣn → PΣn−1, induced by mapping one of the free group generators to the identity, is finitely generated. We show that Kn has cohomological dimension n − 1, and that Hi(Kn;Z) is not finitely generated for 2 ≤ i ≤ n − 1. It follows that Kn is not finitely presentable for n ≥ 3.