Groups with balanced presentations (Q791642)

Let \(G=<x_ 1,...,x_ n\); \(R_ 1,...,R_ m>\) where 2\(\leq m\leq n\). It was shown by \textit{B. Baumslag} and the reviewer [J. Lond. Math. Soc., II. Ser. 17, 425-426 (1978; Zbl 0387.20030)] that if \(n-m\geq 2\) then G has a subgroup of finite index which can be mapped homomorphically onto the free group \(F_ 2\) of rank 2. A similar result holds if \(n-m=1,\) provided one of the \(R_ i's\) is a proper power [\textit{R. Stöhr}, Math. Z. 182, 45-47 (1983; Zbl 0509.20021); \textit{M. Gromov}, Publ. Math., Inst. Hautes Etud. Sci. 56, 5-100 (1982; Zbl 0516.53046)]. In the paper under review the author considers the case \(n-m=0.\) Write each relator \(R_ i\) in the form \(Q_ i^{n_ i}\) where \(Q_ i\) is not a proper power. Reordering the relators if necessary, we can suppose that \(n_ 1\geq n_ 2\geq...\geq n_ m>1.\) Assume \(n_ 2>1\) (i.e. at least two of the relators are proper powers). Let M be the exponent sum matrix of the presentation. Then G has a subgroup of finite index which can be mapped homomorphically onto \(F_ 2\) if either (i) \(\det M\neq 0\) and \((n_ 1,n_ 2,n_ 3,...,n_ m)\neq(2,2,1,...,1),\) or (ii) \(\det M=0\) and there exist \(n_ i\), \(n_ j\) (\(i\neq j)\) with \(hcf(n_ i,n_ j)\geq 1.\) The conditions on the \(n_ i's\) in (i), (ii) cannot be removed in general, as the groups \(<x,y;\quad x^ 2,y^ 2>,\quad<x,y;\quad [x,y]^ 2,[x,y,]^ 3>\) show. For the significance of the above results in a wider context see \textit{M. Edjvet} and \textit{S. J. Pride} [Proc. Groups - Korea 1983, Lect. Notes Math. 1098, 29-54 (1984)].