Higher dimensional isoperimetric functions in hyperbolic groups

Abstract.

We introduce the notion of an $\mathbb{R}$-combing and use it to show that hyperbolic groups satisfy linear isoperimetric inequalities for filling real cycles in each positive dimension. S. Gersten suggested the concept of metabolicity (over $\mathbb{Z}$ or $\mathbb{R}$) for groups which implies hyperbolicity. Metabolicity admits several equivalent definitions: by vanishing of $\ell_\infty$-cohomology, using combings, and others. We prove several criteria for a group to be hyperbolic, $\mathbb{R}$-metabolicity being among them. In particular, a finitely presented group G is hyperbolic iff $H_{(\infty)}^n(G,V)=0$ for any normed vector space V and any $n\ge 2$.