Normal-convexity and equations over groups
An equation over a group \(H\) is of the form \(w=1\) where \(w\in H*F\) for some free group \(F\); free generators of \(F\) are referred to as the unknowns and elements of \(H\) as coefficients. It is common to say that the equation \(w=1\) over \(H\) can be solved over \(H\) is there is a group \(H_ 1\) containing \(H\) and possessing elements which satisfy the equation \(w=1\) when substituted in for the unknowns. The problem of solving such equations can be traced back at least to \textit{B. H. Neumann} [J. Lond. Math. Soc. 18, 4-11 (1943; Zbl 0028.33902)]. The paper under review contributes to the open problem whether every non-singular set of equations (i.e. the matrix of exponent sums has maximal rank) over a group can be solved. While a number of partial answers have recently been obtained for special classes of groups, the approach taken in the paper is to put restrictions on the type of equations while letting the group \(H\) be arbitrary. To this end, a 2-complex \(X\) is said to be Kervaire if all equations over all coefficient groups whose words in the variable letters are the attaching maps of the 2-cells of \(X\) are solvable in an overgroup of the coefficient group. Using the topological notion of normal convexity it is shown that the property of being Kervaire is preserved under subdivision and under certain combinatorial operations on 2-complexes including (the geometric analogues of) amalgamations and HNN extensions. \{Reviewer's remark: The two related papers [\textit{I. M. Chiswell}, \textit{D. J. Collins} and \textit{J. Huebschmann}, Math. Z. 178, 1-36 (1981; Zbl 0443.20030)], [the reviewer, Math. Ann. 258, 17-37 (1981; Zbl 0458.57001)], are not among the references.\}
scientific article; zbMATH DE number 4069183
scientific article