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 pro
perty 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.
We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to exp
lore the relationship between the classical properties (T) and Haagerup and the
We explain how the Transference Principles from Diophantine approximation can be interpreted in terms of geometry of the locally symmetric spaces $T_n=SO(n) backslash SL(n,R) /SL(n,Z)$ with $n>1$, and how, via this dictionary, they become transparent
geometric remarks and can be easily proved. Indeed, a finite family of linear forms is naturally identified to a locally geodesic ray in a space $T_n$ and the way this family is approximated is reflected by the heights at which the ray rises in the cuspidal end. The only difference between the two types of approximation appearing in a Transference Theorem is that the height is measured with respect to different rays in $W$, a Weyl chamber in $T_n$. Thus the Transference Theorem is equivalent to a relation between the Busemann functions of two rays in $W$. This relation is easy to establish on $W$, because restricted to it the two Busemann functions become two linear forms. Since $T_n$ is at finite Hausdorff distance from $W$, the same relation is satisfied up to a bounded perturbation on the whole of $T_n$.
