Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.


If $g\in G$, define its stable translation length as $l(g)=\lim\_n \frac{d(e,g^n)}{n}$.


This number can actually be defined in a more general context: if $G$ acts by isometries on a set $X$, define $l(g)=\lim\_n \frac{d(x,g^n\cdot x)}{n}$ and this do not depend on the point $x$, but we restrict our attention to a word metric in the following.


If $G$ is hyperbolic, then there exists $C\in \mathbb{R}$, such that for every $g\in G$, $l(g)\in C\mathbb{Z}$.


My question is the following: are there examples of groups not satisfying this property for the word metric ? More precisely, fixig a word metric on a group $G$, can we find two elements $g,h\in G$ such that $l(g)$ and $l(h)$ are arbitrarily close ? (settled, see the comment of YCor below).


I am specially interested with hyperbolic elements in relatively hyperbolic groups, so another related question is the following: If $G$ is relatively hyperbolic, can one find two hyperbolic elements $g,h$ such that $l(g)$ and $l(h)$ are arbitrarily close ?


As noticed by YCor, it would be enough to find either a loxodromic element with irrational translation length, or to find a relatively hyperbolic group with loxodromic elements of rational translation length but arbitrarily large denominator.



Rational stable translation length
As already mentioned in the comments, there exist finitely generated groups having non-discrete translation spectra. (There seems to be only a few examples though, it would be interesting to have more.) About relatively hyperbolic groups, we have the following statement:



> 
> **Proposition.** *Let $G$ be a relatively hyperbolic group. There exists some $\epsilon>0$ such that the translation number of every hyperbolic element is a rational $>\epsilon$.*
> 
> 
> 


The fact that the translation number is rational is actually quite general. See in [my other answer here](https://mathoverflow.net/a/429689/122026) for the sketch of a proof that Morse elements in finitely generated groups have rational translation numbers. In order to bound below the translation number, we can consider the action of $G$ on the hyperbolic space $X$ obtained by coning-off $G$ over its parabolic subgroups, i.e. $X$ is the graph obtained from (the Cayley graph of) $G$ by adding an edge between any two vertices that belong to a common parabolic subgroup. The action of $G$ on $X$ is acylindrical and an element of $G$ is loxodromic in $X$ if and only if it is hyperbolic in $G$. Therefore, the translation length in $X$ of a hyperbolic $g \in G$ cannot be arbitrarily small. (See for instance Lemma 2.1 in Fujiwara's article *Subgroups generated by two pseudo-Anosov elements in a mapping class group I*.) But the obvious map $G \to X$ is $1$-Lipschitz, so the translation length of $g$ in $X$ is at most the translation number of $g$ in $G$. The desired conclusion follows.


