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

We study the geometry of infinitely presented groups satisfying the small
cancelation condition C'(1/8), and define a standard decomposition (called the
criss-cross decomposition) for the elements of such groups. We use it to prove
the Rapid Decay property for groups with the stronger small cancelation
property C'(1/10). As a consequence, the Metric Approximation Property holds
for the reduced C*-algebra and for the Fourier algebra of such groups. 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.