Membership problems for positive one-relator groups and one-relation monoids

I will present some recent joint with I. Foniqi and C.-F. Nyberg-Brodda in which we investigate the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. This work is motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis and Meakin (2001). Here a one-relator group is positive if it admits a one-relator presentation where no inverse symbol appears in the defining relator. Such groups were studied by e.g. Baumslag (1971), as well as by Perrin and Schupp (1984) who proved that a one-relator group is positive if and only if it is a one-relation monoid. I will present several new undecidability results, explain how each of them relates to the word problem for one-relation monoids, and outline some of the new methods we developed to prove our results including an approach that involves finding suitable embeddings of certain trace monoids.