Old results and new questions about rapid decay property
I wrote a survey not so long ago. It could not cover all the material. Here is a complement/update.
Let be a finitely generated group. Fix a length function (symmetric, subadditive). On , define the norm as the norm of acting on via its regular representations (i.e. by convolution). Say has rapid decay if for functions with support in the ball of radius , is polynomial in , i.e. .
Equivalently (Connes’ original definition), rapid decay functions belong to the reduced -algebra.
1.1. Case of
Fourier transform maps to , to Laurent polynomials. becomes the sup norm, the reduced -algebra is the algebra of continuous functions, rapid decay functions correspond to smooth functions on the circle.
Cauchy-Schwartz directly gives the linear bound on .
1.2. Applications of RD
Rapid decay has played a role in Lafforgue’s partial solution to Baum-Connes conjecture. Replacing with , which has the same -theory but is much better behaved, helps.
Rapid decay arises in connection with random walks. Let be symmetric. If is nonamenable, convolution powers decay exponentially. Let be the spectral radius of as an operator on . If property RD holds,
Gouezel showed that the sharp asymptotics is . Nevertheless, the RD estimate is much easier.
1.3. Alternative characterizations
Perrone: the norm on balls of coefficients of the regular representation increase polynomially.
Breuillard: the norm, weighted by a power of length, of coefficients, is bounded.
2. Old results
Polynomial growth implies RD. Straightforward.
Jolissaint: conversely, amenability + RD implies polynomial growth. He further showed that , , does not have RD. Also it is transferred from cocompact lattice to an ambient locally cocompact group.
Haagerup: free groups have RD. Indeed, for positive functions with supported in the -ball,
The proof relies on a property of triples of points. Given , let be the median point of . Define if belongs to the geodesic segment , otherwise. Then
A quasification of this proof applies to word hyperbolic groups. The argument applies to a larger class of groups: those which act on coarse median spaces of finite rank (Bowditch). A slight deformation of the proof applies to mapping class groups (Behrstock-Minsky).
It also works for cocompact lattices in , (Ramagga, Robertson, Steger, Lafforgue).
Drutu-Sapir: groups which are hyperbolic relative to subgroups which have RD have property RD as well.
Question (Valette). What about cocompact lattices in , ?
3. Open questions
What about ?
What about acylindrically hyperbolic groups whose point stabilizers have RD? The obvious argument does not work.
3.1. Other representations
In 2014, Breuillard suggested studying RD for other representations. Adrien Boyer has started implementing this. Why would this help?
Theorem 1 (Shalom) Let be a unitary representation. Assume it has a positive vector , i.e. such that
Then dominates the regular representation , i.e. for all ,
It follows that RD for with a positive vector implies RD.
This is what Boyer does: look for representations with positive vectors.
Guentner: does Boyer obtain new obstructions to RD? No, Boyer plays with variants of , but they do not share the KK theory of .
Vaes: is RD qi-invariant? Unclear even for isometries. But existing methods are all large scale.
Wilton: is strongly bolic? Yes, because they are . Lafforgue showed that RD+strongly bolic implies Baum-Connes. So RD is the last obstacle for such groups.