Normal-convexity and equations over groups
type: Item
instance of: scholarly article
title: Normal-convexity and equations over groups
zbMATH Open document ID: 0655.20025
DOI: 10.1007/BF01394345
author: Stephen G. Brick
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143618
review text: 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.\}
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 4069183
zbMATH Keywords: equations over groups
zbMATH Keywords: free products with amalgamations
zbMATH Keywords: generators
zbMATH Keywords: non-singular set of equations
zbMATH Keywords: normal convexity
zbMATH Keywords: 2-complexes
zbMATH Keywords: HNN extensions
reviewed by: J. Huebschmann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4069183