# Example of a group with unsolvable word problem

Today I noticed that the last relator in the 27-relator presentation of a group with unsolvable word problem given in

Donald J. Collins: A simple presentation of a group with unsolvable word problem. Illinois J. Math. 30(1986), no. 2, 230-234.

is a tautology, namely $ka^{-3}ta^3 = ka^{-3}ta^3$. As the total length of the presentation (i.e. sum of the lengths of the relators) is given in the paper as 421, and as the other 26 relators have together length 402, I concluded that the correct last relator would probably have length 19, provided that nothing else is wrong. A Google search for an erratum found a proposed corrected relator $a^{-3}ta^3k = ka^{-3}ta^3$ here (see Section Decision problems). However this relator has length 16, and thus the presentation has then total length 418 rather than 421.

Question: Have I miscounted something, or is there still something more wrong with the presentation in the paper?

• Stefan, it's not a good idea to claim an "error in a classical paper" in the title of the question, so I removed that. – François G. Dorais May 10 '13 at 16:17
• Stefan: Did you consider using the earlier paper "Simple examples of groups with unsolvable word problem" by V.V. Borisov (Math. Notes 6 (1969), 768–775): Borisov has fewer defining relators (only 12). – Misha May 10 '13 at 16:43
• @Misha: Thank you! -- Though Borisov's paper appears to be behind a paywall, which is the reason why I looked at the one by Collins. – Stefan Kohl May 10 '13 at 21:39
• @Francois: I originally formulated the title of the question such as to be as informative as possible. If you think there is no such error in the paper in question, then it would be nice if you would let me know what I have misunderstood. -- Nevertheless I do not insist on the wording of the title. – Stefan Kohl May 10 '13 at 21:56
• The Russian original of Borisov's paper is freely available here: mi.mathnet.ru/eng/mz6959 – Anton Klyachko May 12 '13 at 20:11