Property (RD) for $\mathbb{Q}$

2012-05-21T17:51:55Z

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

Answer by Vahid Shirbisheh for Property (RD) for $\mathbb{Q}$

2012-05-22T03:40:38Z

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).

Property (RD) for $\mathbb{Q}$ - MathOverflow

Property (RD) for $\mathbb{Q}$

Answer by Vahid Shirbisheh for Property (RD) for $\mathbb{Q}$