We introduce the notion of an 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 ℤ or ℝ) for groups which implies hyperbolicity. Metabolicity admits several equivalent definitions: by vanishing of ℓ∞-cohomology, using combings, and others. We prove several criteria for a group to be hyperbolic, ℝ-metabolicity being among them. In particular, a finitely presented group G is hyperbolic iff Hn(∞) (G, V) = 0 for any normed vector space V and any n ≥ 2.
ASJC Scopus subject areas
- General Mathematics