The word problem for two-generator one-relator inverse monoids

Abstract

We study the word problem in two-generator one-relator inverse monoids, proving both decidability and undecidability results. We give the first example of a two-generator one-relator inverse monoid of the form $\mathrm{Inv} \langle a,b \mid w=1 \rangle$ with an undecidable word problem. Furthermore our examples can be chosen to be $E$-unitary. To prove this result we exploit the connection between the word problem for one-relator inverse monoids and the submonoid membership problem for one-relator groups established in work of Ivanov, Margolis, and Meakin (2001). We also give the first example of a two-generator one-relator group, with a reduced word defining relator, with an undecidable prefix membership problem. We investigate the problem of classifying the two-generator one-relator inverse mononids with decidable word problem. For this we prove a general result classifying when a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem. Specifically we prove that a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem if and only if it has decidable rational subset membership problem if and only if the defining automorphism has finite order in the outer automorphism group. We then apply that result to give a sufficient condition for a two-generator one-relator inverse monoid to have decidable word problem in the case its maximal group image is free-by-cyclic of polynomial growth, and we show that that condition can be algorithmically checked. Conversely, we show that when this condition is not satisfied then the two-generator one-relator inverse monoid can be lifted to an example with undecidable word problem and with the same maximal group image (up to taking a free product with the infinite cyclic group).