Normal-convexity and equations over groups

Abstract

Summary. A subgroup S of a group H is said to be normal-convex in H if for any subset R ~ S, the natural map S/((R)) s ~ H/((R))u is injective.In this paper, topological methods are used to show that normal-convexity is preserved under taking free products. In other words, if S is normalconvex in H and if T is normal-convex in K, then S* T is normal-convex m H, K. Similar results are obtained for free products with amalgamation and HNN extensions. The method of proof uses a concept of normal-convexxty 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 group H is of the form w=l where weH,F, F being some free group, with its generators called the unknowns. The elements of H appearlng in w are called the coefficients. The equation w = 1 over H can be solved over H if there is a group H a containing H and possessing elements which satisfy the equation w = 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 group H is changed. Thus different sets of equations may give rise to the same two-complex. We call a twocomplex Kervaire 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 n~-injective subcomplexes. IntroductionThe problem of solving equations over groups can be traced back to the 1940's and the work of B.H. Neumann (see [9]) where the adjunction problem that

Aspherical relative presentations

“…Since the three-cell of Dp(R) is attached along a homologically trivial spherical map, Theorem 4.1 further implies that the composite 7i 2 (K u M (2) ) -» TT 2 (K u M (2) , K x ) ->H 2 (K u M (2) ,K,)…”

“…(This theorem is one of the major results of our paper.) Let P = <H,x; xa l xa 2 xa 3 } where a l ,a 2 …”

“…Finally, note that a special word involving two or more occurrences of B clearly has weight at least 2. (2) In this section we augment the considerations of §3.1 by discussing relative presentations P 2 of the form <//,x; xaxbx~icy where a,b,ceH, and b# 1 ^c. Observe that such presentations are orientable.…”

“…We claim that in any factorization of an element of u* into pieces, each term of the factorization has length 1. For otherwise there would be distinct edges e = (U, 1), f = (V,l) of P st with i(e) = i(/), = *(/)• Then (e/" 1 ) 2 would be an admissible path of length 4, contradicting T(6). It now suffices to show that each element U of u has length at least 6.…”

Branched coverings of 2-complexes and diagrammatic reducibility

“…To state my result let me recall the notion of a Kervaire complex due to S. Brick [20]. A 2-complex A is called "Kervaire" if all equations over all coefficient groups, whose words in the variable letters are the attaching maps of the 2-cells of A, are solvable in an overgroup of the coefficient group.…”