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Igor Dolinka (University of Novi Sad): The word problem for onerelator inverse monoids: New developments 



Wednesday, 4. December 2019, 10:00  12:00


Abstract. In early 1930s, W. Magnus proved his classical result that the word problem is decidable for all onerelator groups (groups given by one defining relation). This result is based on another important theorem proved earlier by Magnus, the Freiheitssatz, which, roughly speaking, locates many free subgroups in onerelator groups. This inspired investigations of the word problem for other algebraic structures defined by a single relation. For example, in the 1960s Shirshov proved that the word problem is decidable for all onerelator Lie algebras. Surprisingly, the problem whether the word problem is decidable for all onerelator monoids is still open, although several important cases have been resolved by Adjan in 1966, and Adyan and Oganessyan in 1987. An important intermediate class of algebraic structures lying between groups and monoids are that of inverse monoids. In 2001 Ivanov, Margolis and Meakin highlighted the importance of investigating onerelator inverse monoids by showing that the (conjectured) decidability of the word problem for onerelator special inverse monoids (in fact, for a particular class of these inverse monoids) would imply a positive solution of the word problem for all onerelator monoids. In this talk, I will present two major recent contributions to this topic: (1) A result of R.D. Gray showing that the word problem for onerelator special inverse monoids is undecidable in its full generality; furthermore, there exists a onerelator group with undecidable submonoid membership problem. (2) The joint results of the speaker and R. D. Gray pertaining to the socalled prefix membership problem for onerelator groups, immediately implying decidability of the word problem for wide classes of onerelator special inverse monoids. Along the way, I will explain the relation of the word problem for onerelator inverse monoids to the word problem of onerelator monoids, and also to several problems in group theory such as the prefix membership problem for onerelator groups and the role of embeddability of rightangled Artin groups. 
Location : Bolyai Intézet, I. emelet, Riesz terem, Aradi vértanúk tere 1., Szeged 
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