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Normal-convexity and equations over groups

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A subgroupS of a groupH is said to be normal-convex inH if for any subsetR⊂S, the natural mapS/《R》 S →H/《R》 H is injective.

In this paper, topological methods are used to show that normal-convexity is preserved under taking free products. In other words, ifS is normal-convex inH and ifT is normal-convex inK, thenS*T is normal-convex inH*K. Similar results are obtained for free products with amalgamation andHNN extensions. The method of proof uses a concept of normal-convexity defined for pairs of topological spaces.

These results and the topological methods are applied to study the question of when a set of equations over a group has a solution in some overgroup. Equations over groups are defined in the following fashion. An equation over a groupH is of the formw=1 wherew∈H*F,F being some free groups, with its generators called theunknowns. The elements ofH appearing inw are called thecoefficients. The equationw=1 overH can be solved overH if there is a groupH 1 containingH and possessing elements which satisfy the equationw=1 when substituted in for the unknowns.

To any set of equations over a group, we associate a two-complex. The manner is analogous to that for presentations. The one-cells correspond to the unknowns, and the two-cells are attached according to the words obtained by ignoring the coefficients. The two-complex so constructed does not change when the coefficients or the groupH is changed. Thus different sets of equations may give rise to the same two-complex. We call a two-complexKervaire if any set of equations associated to it has a solution. Using the topological notion of normal-convexity, we show that the property of being Kervaire is preserved under subdivision, so in particular, it does not depend on the cell structure. Further, we show that the class of Kervaire complexes is closed under combinatorial extensions, connected-sum, cellular two-moves, and amalgamations along two-sided α1-injective subcomplexes.

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  1. Department of Mathematics & Computer Science, University of California, Riverside, 92521, Riverside, CA, USA

    Stephen G. Brick

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  1. Stephen G. Brick
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Brick, S.G. Normal-convexity and equations over groups. Invent Math 94, 81–104 (1988). https://doi.org/10.1007/BF01394345

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