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Abstract. If G is a finitely presented group then its Dehn function-or its isoperimetric inequality-is of interest. For example, G satisfies a linear isoperimetric inequality iff G is negatively curved (or hyperbolic in the sense of Gromov). Also, if G possesses an automatic structure then G satisfies a quadratic isoperimetric inequality.We investigate the effect of certain natural operations on the Dehn function. We consider direct products, taking subgroups of finite index, free products, amalgamations, and HNN extensions.

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

We study the property of tame combability for groups. We show that quasi-isometries preserve this property. We prove that an amalgamation, A *C B, where C is finitely generated, is tame combable iff both A and B are. An analogous result is obtained for HNN extensions. And we show that all one-relator groups are tame combable.

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