Abstract. In early 1930s, W. Magnus proved his classical result that th e word problem is decidable for all one-relator groups (groups given by one defining relation). This result is based on another important theorem prov ed earlier by Magnus, the

An important intermediate class of algebr aic structures lying between groups and monoids are that of inverse monoids . In 2001 Ivanov, Margolis and Meakin highlighted the importance of investi gating one-relator inverse monoids by showing that the (conjectured) decida bility of the word problem for one-relator special inverse monoids (in fact , for a particular class of these inverse monoids) would imply a positive s olution of the word problem for all one-relator monoids. In this talk, I wi ll present two major recent contributions to this topic:

(1) A result of R.D. Gray showing that the word problem for one-relator special inverse mo noids is undecidable in its full generality; furthermore, there exists a on e-relator group with undecidable submonoid membership problem.

(2) The j oint results of the speaker and R. D. Gray pertaining to the so-called pref ix membership problem for one-relator groups, immediately implying decidabi lity of the word problem for wide classes of one-relator special inverse mo noids.

Along the way, I will explain the relation of the word problem fo r one-relator inverse monoids to the word problem of one-relator monoids, a nd also to several problems in group theory such as the prefix membership p roblem for one-relator groups and the role of embeddability of right-angled Artin groups. DTSTAMP:20210927T145734Z DTSTART;TZID=Europe/Budapest:20191204T100000 DTEND;TZID=Europe/Budapest:20191204T120000 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR