On the degree of rapid decay

A finitely generated group $\G$ equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever $a\in\C\G$ is supported on elements of length at most $n$. We show that, for infinite $\G$, the degree $s$ is at least 1/2.


Introduction
Let Γ be a finitely generated group, and fix a word-length on Γ. We say that Γ has property RD if there exist C, s ≥ 0 such that, for all integers n ≥ 0, we have a ≤ C(1 + n) s a 2 whenever a ∈ CΓ is supported on elements of length at most n. Here · denotes the operator norm coming from the regular representation of Γ on ℓ 2 Γ. This property originates from Haagerup's seminal paper [9], where it is shown that free groups have what we now call property RD. The explicit definition of property RD is due to Jolissaint [10]; a result from [10] we would like to quote here is the fact that groups of polynomial growth have property RD. As further examples of groups satisfying property RD, we mention: hyperbolic groups ( [5]) and, more generally, groups that are relatively hyperbolic to subgroups having property RD ( [7]); groups acting freely on finite dimensional CAT(0) cube complexes ( [2]); cocompact lattices in SL 3 (F) for F a local field ( [15]) and for F = R, C ( [11]); mapping class groups ( [1]).
Property RD for a group Γ is relevant to the study of its reduced C * -algebra C * r Γ. The first significant use of property RD is the proof by Connes and Moscovici [4] that hyperbolic groups satisfy the Novikov conjecture. A related K-theoretic application features in the remarkable work of Lafforgue [12], leading eventually to the proof that hyperbolic groups satisfy the Baum-Connes conjecture ( [13], [12]). In another direction, Property RD is used in [8] to show that C * r Γ has stable rank 1 whenever Γ is a torsion-free, non-elementary hyperbolic group. In this note, we are interested in quantifying property RD. For s ≥ 0 consider the following property: (RD s • ) there is C ≥ 0 such that, for all integers n ≥ 0, we have a ≤ C(1 + n) s a 2 whenever a ∈ CΓ is supported on elements of length at most n Note that the satisfaction of (RD s • ) does not depend on the choice of word-length for Γ. Finite groups satisfy (RD 0 • ). Conversely, infinite groups cannot satisfy (RD 0 • ); this was first proved by Rajagopalan in [14]. Rajagopalan's result was a partial positive answer towards the "L p -conjecture" that he formulated around 1960: if G is a locally compact group, and L p (G) is closed under convolution for some p ∈ (1, ∞), then G is a compact. The complete resolution of the L p -conjecture is due to Saeki [16], almost 30 years after its formulation.
The This result is sharp, since (RD s • ) for s = 1 2 is satisfied by all virtually-Z groups. We do not know how to dismiss the main theorem as being trivial, so we settle for the next best thing: an elementary proof (given in Section 3), inspired by Saeki's solution to the L pconjecture. Rajagopalan's proof from [14], using structural results on so-called H * -algebras, does not seem flexible enough to be adaptable here.
It should be pointed out that the main theorem is interesting for infinite torsion groups only, much like the L p -conjecture. Indeed, if Γ contains an infinite cyclic subgroup then the main result is a mere observation (cf. Prop. 2.2 and Prop. 2.3). We are therefore accounting here for the possibility that infinite torsion groups with property RD might exist -a possibility which seems completely open at this time.
Acknowledgments. I thank Gennadi Kasparov and Guoliang Yu for financial support during the summer of 2009. I also thank the referee for some useful comments.

Preliminaries
Throughout this paper, groups are assumed to be finitely generated and equipped with a word-length. The choice of word-length is irrelevant for the discussion herein.

A reformulation of property RD.
An equivalent definition of property RD is the following: a group Γ is said to satisfy property RD if there are C, s ≥ 0 such that a ≤ C a 2,s for all a ∈ CΓ, where Correspondingly, we quantify by considering, for s ≥ 0, the following property: (RD s ) there is C ≥ 0 such that a ≤ C a 2,s for all a ∈ CΓ Informally, (RD s ) is a "de-localized" version of (RD s • ) (recall, the latter is a property defined with reference to balls -hence the label •). The equivalence between the two formulations of property RD is well-known. The next lemma records this equivalence in a precise fashion: The first implication is obvious. The second implication is essentially contained in the proof of Proposition 1.2a) in [11], and works as follows. Assume that Γ satisfies (RD s • ). For n ≥ 0 we let A n be the annulus {g ∈ Γ : 2 n − 1 ≤ |g| < 2 n+1 − 1}. Then for a = a g g ∈ CΓ we have: We conclude that Γ satisfies (RD s+ǫ ) for each ǫ > 0.
In general, (RD s • ) does not imply (RD s ); see the following example.
is the characteristic function of the n-ball of Γ. Since Γ is amenable, we have that a = a 1 for all a ∈ CΓ with positive coefficients; for χ(B n ), we get χ(B n ) = γ(n). It follows that γ(n) ≤ C 2 (1 + n) 2s .
Conversely, assume that γ(n) ≤ Cn 2s for some C > 0. For a ∈ CΓ supported on elements of length at most n, we have a ≤ a 1 ≤ γ(n) a 2 ≤ √ C(1 + n) s a 2 . Thus Γ satisfies (RD s • ). The above fact yields Part (1). Most of Part (2) follows by combining Part (1) with Lemma 2.1; the only thing left to check is that Γ does not satisfy (RD s ) at the critical value s = 1 2 d(Γ). Arguing by contradiction, let us assume that Γ satisfies (RD s ) for s = 1 2 d(Γ). Fix an integer N ≥ 0 and consider where χ(S n ) denotes the characteristic function of the n-sphere of Γ. We have a N ≤ C a N 2,s where C ≥ 0 is independent of N . Since we infer that (1 + n) −d(Γ) |S n | converges. This is absurd, since |S n | n d(Γ)−1 (see [6,VII.32]).

Heredity.
We make the following observation, whose easy proof is left to the reader.
. This proposition also holds for (RD s ) instead of (RD s • ).

Proof of Main Theorem
This section contains the proof of our main result. We start off with two lemmas which are free of any RD assumption. In what follows, products of functions on Γ are convolution products, and inequalities are in the pointwise sense. Proof. The coefficient of each h ∈ B k is at least |B n |, since g −1 h ∈ B n+k whenever g ∈ B n .
For r ≥ 1 and α > 0, consider the following formal sum: Since for each j ≥ 1 we have the desired inequality follows.
We now come to the proof of for all r, k ≥ 1. Fix r ≥ 1 such that χ(B r ) 2 ≥ 2C(1 + r) s -a choice made possible by the fact that the volume growth of Γ is at least linear. Then χ(B r(k+1) ) 2 ≥ 2 χ(B rk ) 2 for all k ≥ 1.
Next, we claim that Z r (α) ∈ ℓ 2 Γ if and only if α > 1 2 . To show this, we compare Z r (α) 2 2 against k −2α . One bound holds in general: For the other bound, we write: The claim is proved.
Pick t such that s < t < 1 2 . Also, pick α, β such that the following are satisfied: and the fact that (RD t ) holds (Lemma 2.1), we deduce that Z r (α) is a bounded operator on ℓ 2 Γ. Then Z r (α)Z r (β) is in ℓ 2 Γ, so Z r (α + β − 1) is in ℓ 2 Γ by Lemma 3.2. This contradiction ends the proof.

Final remarks
According to Lemma 2.1, we have inf{s : Γ satisfies (RD s )} = inf{s : Γ satisfies (RD s • )}; we denote this quantity by rd(Γ) and we call it the RD-degree of Γ. By definition, rd(Γ) is finite precisely when Γ has property RD. Observe that rd(Γ) is independent of the choice of word-length for Γ.
In terms of the RD-degree, our discussion can be summarized as follows: · if Γ is infinite then rd(Γ) ≥ 1 2 ; if Γ is finite then rd(Γ) = 0 · if Γ has polynomial growth then rd(Γ) = 1 2 d(Γ), where d(Γ) denotes the growth degree of Γ · if Γ ′ is a subgroup of Γ then rd(Γ ′ ) ≤ rd(Γ) · if Γ ′ and Γ are commensurable then rd(Γ ′ ) = rd(Γ) These properties suggest that 2 rd(·) can be thought of as a dimension function on finitely generated groups. Computing the RD-degree of other groups would be, of course, interesting in this regard. We single out the following Problem. Compute rd(F r ), where F r denotes the free group of rank r ≥ 2.
Note that the answer is independent of r. It is known that 1 ≤ rd(F r ) ≤ 3 2 (Haagerup's estimates from [9] yield the upper bound; the lower bound is a consequence of Cohen's computations from [3]).
We close by reminding the reader that the following question is open: are there infinite, finitely generated torsion groups which enjoy property RD?