On the Novikov and Boone-Borisov groups (Q1074715)

In his famous pioneering work ''On the algorithmic unsolvability of the word problem in group theory'' [Tr. Mat. Inst. Steklova 44, 1-143 (1955; Zbl 0068.013)] \textit{P. S. Novikov} constructed a finitely presented group N with unsolvable word problem. In [Ann. Math., II. Ser. 70, 207-265 (1959; Zbl 0102.009)], \textit{W. W. Boone} gave another example of such a group. \textit{V. V. Borisov} [in Math. Zametki 6, 521-532 (1969; Zbl 0211.341)] presented a modification of Boone's group \(\Gamma\) (\(\Pi\),P). The aim of this note is to make a survey of the author's recent results on the groups N and \(\Gamma\) (\(\Pi\),P). It is proved that the group N has a ''big'' subgroup \(\tilde N\) with a standard basis. He gives a comparatively short proof for a criterion of equality of words in \(\tilde N.\) It is proved also that the group \(\Gamma\) (\(\Pi\),P) has a standard basis. From these results it is easy to deduce that the word problem in \(\Gamma\) (\(\Pi\),P) is Turing (truth-table) equivalent to the problem of equality to the word P in the initial semigroup \(\Pi\).