A homological characterization of hyperbolic groups (Q1127861)

A finitely presented group \(G\) is hyperbolic if, and only if, both its first unreduced \(\ell_1\)-homology group and its second reduced \(\ell_1\)-homology group with real coefficients vanish. Equivalently, such a group \(G\) is hyperbolic if, and only if, summable 1-cycles (on the universal cover of a \(K(G,1)\) with finite 2-skeleton) can be filled by summable 2-chains and every summable 2-cycle can be approximated by 2-cycles of compact support. A 1-relator group \(G\) is hyperbolic if, and only if, its first \(\ell_1\)-homology group with real coefficients vanishes. If \(\Gamma\) is an arbitrary graph, then every summable 1-cycle with real coefficients can be approximated by 1-cycles of compact support.