Is there any length function on additive group of $\mathbb{Q}$ such that $\mathbb{Q}$ is of polynomial growth WRT this length function? What about the multiplicative group of $\mathbb{Q}$ instead?



growth of infinitely generated groups
Yes for $\mathbf{Q}$, no for $\mathbf{Q}^\*$.


For $\mathbf{Q}$, write it as the union of an increasing sequence $L\_n$ with $L\_1=\mathbf{Z}$ and $L\_n$ of finite index over $L\_1$. Pick a function $F$ with fast growth and define $l'(r)=|r|+F(\sup\{n:r\notin L\_n\})$.


For $\mathbf{Q}^\*$, it contains a subgroup isomorphic to $\mathbf{Z}^d$ for every $d$, so for every length the growth is at least polynomial of degree $d$. So it can't be polynomial.


Added-1: for $\mathbf{Q}$ we can arrange so that $[L\_n:L\_1]\le n$ for all $n$. Then we can pick $F$ to be the identity and then the growth is at most quadratic. 


Added-2: the argument extends, showing that an abelian group admits a length with polynomial growth iff it's countable and has finite $\mathbf{Q}$-rank. 


