The Rapid Decay Property

Let $G$ be a countable group with a length function $L$. That means $L(1)=1, L(gh)\le L(g)+L(h), L(g^{-1})=L(g)$. For example, a finitely generated group and its word length. Property Rapid Decay (introduced by Haagerup and Jolissaint) says that the operator norm of an element of the group algebra $\mathbb{C}G$ is not much bigger than its $l_2$-norm. The property is interesting because it is related to the Baum-Connes conjecture, and is also useful for studying random walks on groups. Also many groups, including some very interesting groups satisfy this property: all Gromov hyperbolic groups, mapping class groups, Artin groups of large type, groups acting nicely on CAT(0)-cubical complexes, etc. There are several reformulations of the property RD, one of the easiest reformulations was used in our paper with Cornelia Drutu.

Let $\phi, \psi$ be two elements in $\mathbb{R}G$, all elements in the support of $\phi$ have length $r$. Then RD is equivalent to the existence of a universal polynomial $P(r)$ such that $|\phi\psi|^2\le P(r)|\phi|^2|\psi^2|$ where $|.|$ is the $l_2-$norm. It is easy to see that $\phi\psi$ is a function $\gamma$ such that $\gamma(k)=\sum\limits_g \phi(g)\psi(g^{-1} k)$ for every $k\in G$. Since the norm squared is the sum of squares of coefficients, it is clear that the problem is about triangles with side labels $(g,h,gh)$ in the Cayley graph of $G$: the “more” triangles with a given base label, the harder it is to get RD.

Even though the known classes of groups with RD are quite diverse, the methods of proving RD are “asymptotically similar”.  The reason why the free groups have RD is that every geodesic triangle on a tree has a center which belongs to every side of the triangle. For Cayley graphs of hyperbolic groups, a center of a triangle may not belong to all three sides, but it is at bounded distance from all three sides (this is Rips’ definition of hyperbolic groups). For triangles in the Cayley graphs of relatively hyperbolic groups (as in our paper with Cornelia Drutu) and in symmetric spaces of Lie groups such as  $SL_3(\mathbb{R})$ every triangle has an “inscribed” nice and relatively small triangle from a certain family of triangles. Indira Chatterji and Kim Ruane used clouds of centers, and Laura Ciobanu, Derek Holt and Sarah Rees used a condition which can be interpreted as a center-like condition.

I just wrote a paper “The Rapid Decay property and centroids in groups”, where an easy to formulate and check “centroid” condition is given. It follows from the centroid-type conditions used before and implies property RD. A not quite successful  attempt to do that was made at the end of our paper with Cornelia Drutu. There we formulated our property “(**)-relative hyperbolicity”. But that property is not general enough and  the definition of (**) in our paper contains errors.

The centroid property is the following.

Let $G$ be a countable group acting almost freely (i.e., the stabilizers of points have uniformly bounded finite orders) by isometries on a metric space $(X, \mathrm{dist})$,  $x_0\in X$. We assume that $L$ is the length function defined by $L(g)=\mathrm{dist}(x_0,g\cdot x_0)$. Let $\mathfrak{c}$ be a map from the set of pairs $G^2=G\times G$ to $X$. We can view $G$ as embedded into $X$ (by the map $g\to g\cdot x_0$), a pair $(g,k)\in G\times G$ as the vertices of triangle $(x_0,g\cdot x_0,k\cdot x_0)$, and $c=\mathfrak{c}(g,k)$ as a “center” of that triangle. We say that $G$ and $\mathfrak{c}$ satisfies the  centroid property  if for some polynomial $P(x)$ we have

$(c_1)$ For every $k\in G$ and every $r>0$ the number of elements $\mathfrak{c}(g,k), L(g)\le r,$ does not exceed $P(r).$

$(c_2)$  For every $g$  in $G$ the number of elements $\mathfrak{c}(g,k)$, $k$ in $G,$ does not exceed $P(L(g))$.

$(c_3)$ For every$h\in G$  the number of elements in the set $\{g^{-1}\mathfrak{c}(g,gh), L(g)\le r\}$ does not exceed $P(r).$
In this case $X$ is called the space of centroids of $G$ and $\mathfrak{c}$ is called the centroid map.

The meaning is clear: if we fix the label (in $G$) of one side of the triangle with vertices  $(1,g,k)$ and vary the third vertex keeping the length of one side $r$ , then the number of possible centroids of these triangles is not large (at most a polynomial in $r$).

In the paper, I show that almost all groups that are known to have RD such as hyperbolic groups, many uniform lattices in direct products of semisimple Lie groups of ranks 1 and 2, the mapping class groups, infinitely presented small cancelation groups, Artin groups of large type, have the centroid property or its relative version. The only  class for which it is not known (to me) are  graph products of groups. An undergraduate student Mitch Kleban  is now working on that, it is his Summer REU problem.  I think that the centroid property is interesting in its own right as a very weak form of hyperbolicity.

Another thing which I included in the paper is an easy non-amenability-like corollary from RD. If in the definition of RD one restricts oneself to indicator functions of finite sets, then it is easy (one line) to prove that for every group with RD there is a polynomial $P(r)$ such that for every two finite sets of elements $S, X$ where all elements of $S$ have length $\le r$, we have

$(1) \hskip 2 in |SX|\ge \frac{|S||X|}{P(r)}.$

In particular, it immediately implies that $G$ does not have amenable subgroups of superpolynomial growth. It would be interesting to find out if (1) (which I call “superpolynomial expansion property) implies RD. Using (1), it is easy to observe that the free product of all free Abelian groups of finite ranks with an appropriate length function is an example of a countable group without RD and without amenable subgroups of superpolynomial growth. Alexander Olshanskii proved that one of the standard embeddings of countable groups into 2-generated groups does not add extra amenable subgroups (i.e. every amenable subgroup of the bigger group is either cyclic of is conjugated to a subgroup of the smaller group), and cyclic subgroups in the bigger group that are not conjugated to subgroups of the smaller group are undistorted. Therefore there exists a 2-generated group without RD and without amenable subgroups of superpolynomial growth. Another example was constructed by Denis Osin. It would be interesting to find a finitely presented example. It should not be that difficult using S-machines and one of the numerous versions of the Higman embedding theorem.

A very interestiong problem (obviously related to a problem of Alain Valette whether uniform lattices in higher rank semi-simple Lie groups always have RD) is whether uniform lattices in $SL_4(\mathbb{R})$ satisfy the (relative) centroid property or the superpolynomial expansion property.