Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

Abstract

We study the geometry of infinitely presented groups satisfying the small cancelation condition $ C'\''(1/8)$ , and define a standard decomposition (called the \dec decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups $ G$ with the stronger small cancelation property $ C'\''(1/10)$ . As a consequence, the Metric Approximation Property holds for the reduced $ C^*$ --algebra $ C^*_r(G)$ and for the Fourier algebra $ A(G)$ of the group $ G$ . Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree $ 2$ has a basis of power continuum. The present work can be viewed as a first non-trivial step towards a systematic investigation of direct limits of hyperbolic groups.