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On irreducible cyclic presentations of the trivial group. (English) Zbl 1307.20029

Let \(w=w(x_0,\ldots,x_{n-1})\) be a freely and cyclically reduced element of the free group \(F=F(x_0,\ldots,x_{n-1})\) and let \(\theta\colon x_i\to x_{i+1}\pmod n\) be an automorphism of \(F\). Then \(G_n(w)=\langle x_0,\ldots,x_{n-1}\mid w,w\theta,\ldots,w\theta^{n-1}\rangle\) is called a cyclically presented group. This presentation for \(G_n(w)\) is irreducible if the group does not decompose as a free product of copies of another cyclically presented group in a straightforward way. This article is the latest in a series of papers that investigate for which \(n\) and \(w\) the presentation \(G_n(w)\) is an irreducible presentation of the trivial group. (One motivation for research into this question is that any such presentation can serve as a test case for the Andrews-Curtis conjecture.)
The split extension of \(G_n(w)\) by \(\mathbb Z_n\), induced by \(\theta\), has a presentation \(H=\langle x,t\mid t^n,\;w(x,t)\rangle\) where \(w(x,t)\) is obtained from \(w\) by the rewrite \(x_i\mapsto t^{-i}xt^i\). Previous investigations have led to the formulation of the following conjecture: If \(G_n(w)\) is an irreducible cyclically presented group whose corresponding word \(w(x,t)\) has length \(l\leq 2n\) then the group \(G_n(w)\) is non-trivial.
The article [Int. J. Algebra Comput. 20, No. 3, 417-435 (2010; Zbl 1234.20039)] by J. E. Cremona and M. Edjvet obtained a list of all possible test words \(w(x,t)\) of (free product) length \(l\leq 15\), which was used to show that if \(l\leq 15\) and \(6\leq n\leq 100\) then \(G_n(w)\) cannot be irreducible and trivial. The present article extends that experiment to produce all test words of length \(l\leq 20\). This new list is used to confirm the conjecture for \(l\leq 17\) and \(n\leq 100\). The more detailed results are also presented, namely for \(l=16,17\) a list of the 54 irreducible cyclic presentations of the trivial group that were obtained, and a list of the 24 test words \(w(x,t)\) with \(l=16,17\) for which triviality of \(G_n(w)\) is undecided for \(4\leq n\leq 100\).
The determination of all possible test words \(w(x,t)\) with \(l\leq 20\) involves computer searches up to a bound on \(n\), combined with number theoretic arguments to show that no further test words occur for higher \(n\) values. Once the test words are obtained, group theoretic techniques are applied to establish triviality or otherwise of the corresponding groups \(G_n(w)\). This can involve KBMAG or MAF to obtain the order of \(G\), or MAGMA to find non-trivial quotients or low index subgroups. In certain cases, embedding theorems from the theory of cyclically presented groups are used.

MSC:

20F05 Generators, relations, and presentations of groups
57M20 Two-dimensional complexes (manifolds) (MSC2010)
20-04 Software, source code, etc. for problems pertaining to group theory
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 1234.20039

Software:

kbmag; Magma; MAF
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Full Text: DOI

References:

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