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Question 1. Is it known that for some free Burnside groups the word problem is undecidable?

Provided that the answer is negative, what about the following easier question.

Question 2. Is there a known example of a finitely generated (and preferably finitely presented) group $G$ and an integer $k$ such that all elements of $G$ have order at most $k$ and the word problem in $G$ is undecidable?

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I am far from being an expert, but are there any examples of infinite finitely presented torsion groups? – Yiftach Barnea Jun 2 '11 at 11:38
@Yiftach: I've just googled that this was an open question in 2002 (paragraph 3, page 3 in the article "non-amenable finitely presented torsion-by cyclic groups" by Ol'shanskii and Sapir), so perhaps such examples are still not known. – Łukasz Grabowski Jun 2 '11 at 11:50
@Yiftach: there's also an article of M. Sapir from 2007 "Some group theory problems", where he mentions existence of finitely presented infinite torsion groups as an open problem. – Łukasz Grabowski Jun 2 '11 at 13:22
I asked Martin Bridson in January 2010, at which point the question was still open. I have no reason to think this has changed. – Jonathan Kiehlmann Jun 20 '11 at 12:39
up vote 9 down vote accepted

Concerning question 1: for free Burnside groups of odd exponent $n\geq 665$ the decidability was shown by S.I.Adian. Lysionok proved the same in the case of even exponent $n=16k\geq 8000$. The corresponding deciding procedure is just Dehn's algorithm.

It will be interesting to know how to solve the word problem for groups $B(m,n)$ when $n=2k$, where $k\geqslant 665$ and odd. These groups are infinite (it follows from the cited result of Adian), but it seems that the decidability of the word problem for these groups is an open question.

As to the question 2:

The pure existence of non-finitely presented group with undecidable w.p. easily follows from the result of S.I.Adian who proved that there are continuum non-isomorphic periodic groups with fixed number of generators $m\geqslant 2$ satisfying the periodic law $X^n=1$ if $n$ is odd and $n \geq 665$. One should mention, that the number of "additional relations" (besides all periodic) is not necessarily finite.

Later M.Sapir constructed an example of periodic group with finite number of additional relations. The paper is "On the word problem in periodic group varieties", IJAC, 1991, V.1, No.1.

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Many thanks, this answers all the questions I had, even the one which I didn't ask explicetely, i.e. whether there exists a periodic group with recursive presentation and undecidable word problem. – Łukasz Grabowski Jun 4 '11 at 16:37
The word problem was shown to be solvable for sufficiently large exponents by Ivanov. – Ian Agol Oct 1 '15 at 13:38

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