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

Arzhantseva, G
Drutu, C

21 August 2018

## Last Updated:

2020-10-26T02:17:48.753+00:00

## DOI:

10.4153/CJM-2018-036-7

## abstract:

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.

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