Cohomological lower bounds for isoperimetric functions on groups
In previous papers [see \textit{S. M. Gersten}, in Combinatorial and geometric group theory, Lond. Math. Soc. Lect. Note Ser. 224, 87-104 (1995; Zbl 0851.20036)], the author has worked out an \(\ell_\infty\)-cohomology which is adapted to the study of isoperimetric functions on finitely presented groups. If a finitely generated group \(G\) possesses an Eilenberg-MacLane space \(X\) of type \(K(G,1)\), where \(X\) is a CW-complex with a finite \(n\)-skeleton \(X^{(n)}\), then \(H_{(\infty)}^i(G)\) is defined for \(i\leq n\) by taking uniformly bounded cellular cocycles in the universal cover \(\widetilde X\), modulo the image under coboundary of uniformly bounded cochains. The groups \(H_{(\infty)}^i(G)\) are quasi-isometry invariants. A class in \(H_{(\infty)}^2(G)\) gives a lower bound for the Dehn function of a finite presentation for \(G\) (the Dehn function is the minimal isoperimetric function for the presentation). If the cohomology class is zero, this lower bound is linear. There is a Mayer-Vietoris exact sequence for \(\ell_\infty\)-cohomology which is associated to a splitting of groups. If \(G\) splits over a finitely presented subgroup, then the author constructs classes in \(H_{(\infty)}^2(G)\) which reflect the splitting and which serve as lower bounds for isoperimetric functions. He proves that \(H_{(\infty)}^2(G)=0\) if \(G\) is word hyperbolic and he presents results on the converse of this theorem. These results concern the Bestvina-Feighn combination theorem for word hyperbolic groups. The author gives several theorems on the non-vanishing of the \(\ell_\infty\)-cohomology. He introduces the notion of metabolic groups, as finitely presented groups such that \(H_{(\infty)}^2(G,A)=0\) for all normed Abelian coefficient groups \(A\). He shows that such groups \(G\) satisfy the linear isoperimetric inequality for cycles in all dimensions and that all \(\ell_\infty\)-cohomology groups for all coefficients in degrees at least 2 vanish. The corresponding assertions for arbitrary hyperbolic groups are open questions. The authors proves in fact that metabolic groups are hyperbolic, and he asks whether the converse is true.
scientific article; zbMATH DE number 1307791