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Let $G$ be a (multiplicative) group with neutral element $e$. Let us call a function $\ell: G \rightarrow \mathbb{N}$ satisfying

  1. $\ell(e)=0$;
  2. $\ell(x^{-1})=\ell(x)$ for all $x\in G$;
  3. $\ell(xy) \leq \ell(x)+\ell(y)$ for all $x,y \in G$;

a length function on $G$. It is called proper if for every $n \in \mathbb{N}$ the set $\{ x \in G \mid \ell(x)=n\}$ is finite.

As I understand, in general (proper) length functions on groups can look very wild. I was wondering if for very simple groups, e.g. the integers $\mathbb{Z}$ more can be said?

Question: Is it possible to classify all (proper) length functions on $\mathbb{Z}$? What about $\mathbb{Z}^n$, $n \in \mathbb{N}$?

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    $\begingroup$ If $f$ is a concave function mapping 0 to 0 and tending to $\infty$, then $f\circ\ell$ is a length function as soon as $\ell$ is a length function. This makes basically impossible to classify all length functions. (For instance there are sublinear length function growing as fast as any prescribed sublinear function. Also, there are proper length function growing slower than any prescribed function tending to $\infty$.) $\endgroup$
    – YCor
    Commented Jul 29, 2022 at 12:10

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