Notes of Indira Chatterji’s Cambridge workshop lecture 12-05-2017

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 ${G}$ be a finitely generated group. Fix a length function (symmetric, subadditive). On ${{\mathbb C} G}$, define the norm ${\|f\|_*}$ as the norm of ${f}$ acting on ${\ell^2 G}$ via its regular representations (i.e. by convolution). Say ${G}$ has rapid decay if for functions with support in the ball of radius ${R}$, ${\|f\|_*}$ is polynomial in ${R}$, i.e. ${\|f\|_*\leq C\,R^D\|f\|_2}$.

Equivalently (Connes’ original definition), rapid decay functions ${H^\infty}$ belong to the reduced ${C^*}$-algebra.

1. Motivation

1.1. Case of ${{\mathbb Z}}$

Fourier transform maps ${\ell^2({\mathbb Z})}$ to ${L^2(S^1)}$, ${{\mathbb C}{\mathbb Z}}$ to Laurent polynomials. ${\|f\|_*}$ becomes the sup norm, the reduced ${C^*}$-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 ${\|f\|_*}$.

1.2. Applications of RD

Rapid decay has played a role in Lafforgue’s partial solution to Baum-Connes conjecture. Replacing ${C_r}$ with ${H^\infty}$, which has the same ${KK}$-theory but is much better behaved, helps.

Rapid decay arises in connection with random walks. Let ${f\in{\mathbb C} G}$ be symmetric. If ${G}$ is nonamenable, convolution powers ${f^{(2n)(e)}}$ decay exponentially. Let ${\rho_f}$ be the spectral radius of ${f}$ as an operator on ${\ell^2 G}$. If property RD holds,

$\displaystyle \begin{array}{rcl} c\,n^{-2D}\leq \rho_f^{-2n}f^{(2n)}(e)\leq C\,n^{-1}. \end{array}$

Gouezel showed that the sharp asymptotics is ${n^{-3/2}}$. Nevertheless, the RD estimate is much easier.

1.3. Alternative characterizations

Perrone: the ${\ell^2}$ norm on balls of coefficients of the regular representation increase polynomially.

Breuillard: the ${\ell^2}$ 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 ${Sl(n,{\mathbb Z})}$, ${n\geq 3}$, 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 ${f,g,h}$ with ${f}$ supported in the ${R}$-ball,

$\displaystyle \begin{array}{rcl} (f\star g\star h)(e)\leq C\,R^D \|f\|_2\|g\|_2\|h\|_2. \end{array}$

The proof relies on a property of triples of points. Given ${x,y\in G}$, let ${t}$ be the median point of ${x,y,y^{-1}x^{-1}}$. Define ${\langle T_f \delta_a,\delta_b\rangle=f(a^{-1}b)}$ if ${e}$ belongs to the geodesic segment ${[a,b]}$, ${=0}$ otherwise. Then

$\displaystyle \begin{array}{rcl} (f\star g\star h)(e)=trace(T_f T_g T_h)\leq \|f\|_{HS}\|g\|_{HS}\|h\|_{HS}\leq \sqrt{R}\|f\|_2. \end{array}$

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 ${Sl(3,K)}$, ${K={\mathbb Q}_p,{\mathbb R},{\mathbb C},\mathbb{H}}$ (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 ${Sl(n,K)}$, ${n>3}$?

3. Open questions

What about ${Out(F_n)}$?

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 ${\pi:G\rightarrow U(H)}$ be a unitary representation. Assume it has a positive vector ${\xi\in H}$, i.e. such that

$\displaystyle \begin{array}{rcl} \langle\pi(g)\xi,\xi\rangle\geq 0 \quad \textrm{for all }g\in G. \end{array}$

Then ${\pi}$ dominates the regular representation ${\lambda}$, i.e. for all ${f\in{\mathbb C} G}$,

$\displaystyle \begin{array}{rcl} \|\pi(f)\|\geq\|\lambda(f)\|. \end{array}$

It follows that RD for ${\pi}$ with a positive vector implies RD.

This is what Boyer does: look for representations with positive vectors.

4. Questions

Guentner: does Boyer obtain new obstructions to RD? No, Boyer plays with variants of ${H^\infty}$, but they do not share the KK theory of ${C_r}$.

Vaes: is RD qi-invariant? Unclear even for isometries. But existing methods are all large scale.

Wilton: is ${Sl(n,K)}$ strongly bolic? Yes, because they are ${CAT(0)}$. Lafforgue showed that RD+strongly bolic implies Baum-Connes. So RD is the last obstacle for such groups.