Generators increasing the word metric

Question

Let $G$ be an infinite group with a finite generating set $T$ which is symmetric ($T=T^{-1}$) and let $|\cdot|$ be the corresponding word metric, i.e. $|g|$ is the minimal number of (not necessarily distinct) elements from $T$ one needs to multiply to get $g$.

Is it true that for every $g\in G$ there is a $t\in T$ such that $|gt|>|g|$?

Answer 1

Consider $G=\mathbb{Z}$ with $T=\{\pm2,\pm3\}$. Then $|1|=2$, and $|1+t|\leq 2$ for all $t\in T$.