Finiteness properties for a subgroup of the pure symmetric automorphism group
Abstract
Let Fn be the free group on n generators, and Pn be the group of automorphisms of Fn which send each generator to a conjugate of itself. Let Kn be the kernel of the homomorphism from Pn to Pn-1 induced by mapping one of the free group generators to the identity. We show that Kn has cohomological dimension n-1, and that the ith cohomology groups are infinitely generated for all i between 2 and n-1. It follows that Kn is not finitely presentable for n>2.
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