Seminar 64. Undecidability of the word problem for one-relator inverse semigroups

Inverse semigroups are an intermediate structure between groups and semigroups, and can be used to model partial symmetries. They exhibit much exotic behaviour; for example, while the word problem for free inverse semigroups is decidable, it is somewhat non-trivial to solve, quite unlike the case of free groups and semigroups. In recent years, presentations of inverse semigroups have been investigated, and several surprising results have started appearing. Magnus proved already in 1932 that one-relator groups have decidable word problem; Adian proved in 1960 that certain one-relation semigroups have decidable word problem (although the problem remains open in general). By contrast, Gray proved in 2020 that there exist one-relation inverse semigroups with undecidable word problem. This result, and recent extensions of it, will be the focus of the talk. I will give an introduction to the theory of inverse semigroups, and an overview of free inverse semigroups and the solution to their word problem via Munn trees. I will present some results about presentations of inverse semigroups and the basic tools available. Finally, I will present the proof of Gray's result, which passes via submonoids of right-angled Artin groups, and some recent extensions of it to positive two-relator inverse monoids.