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Do the additive group or the multiplicative group of $\mathbb{Q}$ have property (RD) (Rapid Decay)?

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    And what's property (RD)? – Felix Goldberg May 21 '12 at 18:12
  • Surely yes for any abelian group, by Cauchy-Schwarz? Unless there is some subtlety in the definition of length function that I have missed... – Yemon Choi May 21 '12 at 18:13
  • @Felix: there is a nice exposition at the start of I. Chaterji's thesis http://www.math.ethz.ch/u/burger/chaterji.pdf – Yemon Choi May 21 '12 at 18:14
  • @Felix Goldberg: probably Rapid Decay. – Lee Mosher May 21 '12 at 18:22
  • @Vahid: you might want to read the FAQ about how to formulate questions better on MO. You will greatly improve the quality of your reader's feedback that way. – Lee Mosher May 21 '12 at 18:23
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    Property (RD) depends on the metric. For example, a cyclic group with exponentially distorted metric does not have property (RD) (as any amenable group of exponential growth). So the answer depends on the metric you choose. For a finitely generated group, the metric is usually the standard word metric, but in your case the groups are not finitely generated. –  May 21 '12 at 19:30
  • Thanks for comments. Yes I mean property Rapid decay. And as Mark noted it depends on the length function and the growth associated to the length function. It brings us to the question "can we define a length function over $\mathbb{Q}$ such that $\mathbb{Q}" is of polynomial growth wrt this length function?". –  May 21 '12 at 19:37
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    Please use the "edit" link to refine your question. – S. Carnahan May 22 '12 at 01:05

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Thanks to 'Yves Cornulier's answer to my other question about the growth of $\mathbb{Q}$, we now know (1) there is a length function on the additive group of $\mathbb{Q}$ which makes $\mathbb{Q}$ of polynomial growth. (2) there is no length function on $\mathbb{Q}^\times$ making it of polynomial growth.

We can modify a theorem by Jolissaint which says: if $G$ is an amenable (finitely generated) group, then $G$ has (RD) if and only if $G$ is of polynomial growth. To generalize this theorem to infinitely generated groups one only needs to show that if $G$ has (RD) w.r.t. some length function $L$ then $\{ g\in G; L(g)\leq r\}$ is finite for all $r\geq 0$. This is easily done by introducing to sequence of functions in $\mathbb{C}G$ (I will give details in the next few days).

Now, since $\mathbb{Q}$ and $\mathbb{Q}^\times$ are both amenable, $\mathbb{Q}^\times$ does note have (RD) and $\mathbb{Q}$ has (RD).