Finiteness properties for a subgroup of the pure symmetric automorphism group

Abstract

Let F_n be the free group on n generators, and PΣ_n be the group of automorphisms of F_n which send each generator to a conjugate of itself. Let K_n be the kernel of the homomorphism from PΣ_n to PΣ_{n-1} induced by mapping one of the free group generators to the identity. We show that K_n 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 K_n is not finitely presentable for n>2.

Originally titled "Finiteness properties for the kernel of pure motions of n unlinked loops"; error in Lemma 3.2 removed; argument of main theorem simplified

References in corpus (1)

• The integral cohomology of the group of loops