Abstract: The Sigma-invariants of a finitely generated group are a nested sequence of geometric invariants which determine the finiteness properties of sufficiently large (namely, co-abelian) subgroups. The Sigma-invariants are notoriously hard to compute, but miraculously Ken Brown found an algorithmic way to compute them from a group presentation that has only one relator. Come, learn how to tell if subgroups of your favorite one-relator groups are finitely generated!