Suppose we have a group $G$ which is finitely generated , and let $|\cdot |$ denote some word metric on it. Must there be an element $a\in G$ such that $|a^n|\ge c\cdot n$ for some $c>0$?


My intuition says that the answer is "yes", but I wasn't able to find a proof in the general case. There are many cases that I know that the answer is yes - abelian and nilpotent groups, free products, Baumslag-solitar subgroups and many more.


**What I managed to do** - If there is a finite index subgroup $H$ of $G$ such that the abelianization $\frac{H}{[H,H]}$ is infinite, then one can find such an element. 


The reason that this is true is not so hard - by quasi-isometry, one might as well ask the question for H. Now, we know that $\frac{H}{[H,H]}$ is a finitely generated abelian group (as $H$ is, beacuse $G$ is). Therefore it must have a further quotient which is isomorphic to $\mathbb{Z}^d$. We let $a\in G$ be some element of $G$ which does not vanish after projecting it to $\mathbb{Z}^d$, and let $v$ be its image there


Suppose $|\cdot|$ is a word metric which corresponds to $S \subset G$. Let $T$ be the image of $S$ in $\mathbb{Z}^d$, and denote the associated word metric by $|\cdot |'$. Then it's clear that:


$$
|a^n| \ge |n\cdot v|' \ge \frac{||n\cdot v||}{\max\_{t\in T}{||t||}}=c\_{v,T}\cdot n
$$


for any norm $||\cdot ||$ on $\mathbb{R}^d$. This gives the linear lower bound we wanted.


***Edit***
I forgot to mention that $G$ is not the trivial group or a torsion group. Thank you Derek Holt for mentioning it!



Must the powers of some element always grow linearly with respect to a word metric?
(Probably the question would be more suitable for MathSE)


The answer is no. Well, the trivial group is a counterexample. Also finite groups are counterexamples. So first and for all you should have specified that you assume the group to be infinite. But then the answer is still no.


In a finitely generated group, an element $g$ is called *distorted* if $\lim |g^n|\_S/n=0$ (the limit exists, and that the limit vanishes does not depend on the choice of finite generating subset $S$). So you're asking about the existence of an undistorted elements.


Torsion elements are distorted. Hence any finitely generated torsion group have all its elements distorted, and many infinite examples are known (Golod-Shafarevich, Alëshin-Grigorchuk, Burnside groups by work of Adian, etc).


There also exist infinite torsion-free finitely generated groups with all elements distorted: it includes groups with finitely many conjugacy classes constructed by Osin (and maybe Ivanov before but I don't remember if his examples are torsion-free).


There are positive results for nice (restricted) classes of groups: for instance any infinite finitely generated group that is either elementary amenable (e.g., solvable), Gromov-hyperbolic, or has a faithful finite-dimensional representation over some field, has an undistorted element.


