Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ denotes the word length of $g \in G$.

M. Coornaert proved that the growth of $W_n$ is purely exponential i.e there exists $\lambda, C>1$ such that $$\frac{1}{C}\lambda^n \le \#W_n \le C \lambda^n$$ for all $n \in \mathbb{Z}_{\ge 0}$ and where $\#W_n$ denotes the cardinality of $W_n$.

Fix $M \in \mathbb{R}$. I'm interested in the growth rate of the quantity $$\#(\varphi^{-1}([-M,M]) \cap W_n)$$ and in particular how this compares to the growth rate of $\#W_n$. It seems plausible to me that the above quanitity takes up an asymptotically vanishing proportion of $\#W_n$ i.e $$ \frac{\#(\varphi^{-1}([-M,M]) \cap W_n)}{\#W_n} \rightarrow 0.$$

However, I am unable to prove this or to provide a counterexample. Is anything known about this problem?

Any thoughts and/or suggestions would be greatly appreciated. Many thanks.

annulardensity is zero and one may try to generalize this to non-elementary hyperbolic groups. Now, what if $G$ is freely generated by $a$ and $b$ and $\varphi(a) = 1$ and $\varphi(b) = \sqrt{2}$? Is the picture already clear for $G = \mathbb{Z}^2$? $\endgroup$ – Luc Guyot Oct 4 '18 at 21:11