Fibonacci type presentations and 3-manifolds (Q347258)

Let \(F_n=\langle x_0,\ldots ,x_{n-1}\rangle\) be the free group of rank \(n\) and let \(w\) be a word in \(F_n\). Let \(\phi: F_n\to F_n\) be the automorphism defined by \(x_i\to x_{i+1}\) (subscripts mod \(n\)) and NEWLINE\[NEWLINE{\mathcal G}_n(w)=\langle x_0,\ldots ,x_{n-1}\mid \phi^i(w), (0\leq i\leq n-1)\rangle .NEWLINE\]NEWLINE The corresponding group \(G_n(w)\) is called a \textit{cyclically presented group}. The authors study the presentations \({\mathcal G}_n(m,k)={\mathcal G}_n(x_0x_mx_k^{-1})\) (\(n\geq 1\), \(0\leq m,k\leq n-1\)) and the groups \(G_n(m,k)\) they define, the so called \textit{groups of Fibonacci type}. There are also presentations \({\mathcal H}(n,m)={\mathcal G}_n(m,1)\) studied in the literature.NEWLINENEWLINEIn this paper, the questions which presentations \({\mathcal G}_n(m,k)\) are presentations of spines of 3-manifolds and which groups \(G_n(m,k)\) occur as fundamental groups of closed 3-manifolds are covered. The answer to the second question is given with the exception of the groups \(H(9,4)\) and \(H(9,7)\). The first question is completely answered with the help of the first question.NEWLINENEWLINEIt turns out, that the only cyclically presented groups of type \({\mathcal G}_n(m,k)\) which are fundamental groups of 3-manifolds are Fibonacci, Sieradski and cyclic groups. \({\mathcal G}_n(1,2)\) is a presentation of a Fibonacci group and \({\mathcal G}_n(2,1)\) of a Sieradski group.