Abstract:
Rapid decay property (property (RD)) for groups, generalizes
Haagerup’s inequality for free groups and so for example of free groups have
property RD. Property RD provides estimates for the operator norm of those
functions (in the left-regular representation) in terms of the Sobolev norm.
Even more, property RD is the noncommutative analogue of the fact that
smooth functions are continuous. This property RD for groups has deep
implications for the analytical, topological and geometric aspects of groups.
It has been proved that groups of polynomial growth and classical hyperbolic
groups have property RD, and the only amenable discrete groups that have
property RD are groups of polynomial growth. He also showed that many
groups, for instance 𝑆𝐿3(ℤ), do not have the Rapid Decay property.
Examples of RD groups include group acting on CAT(0)-cube complexes,
hyperbolic groups of Gromov, Coxeter groups, and torus knot groups. The
symmetry group of a tiling pattern of the plane is called a crystallographic
group. The discrete Heisenberg group is the multiplicative group Η3 of all
matrices of the form
