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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.

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    $\begingroup$ If $\varphi(G)$ is isomorphic to $\mathbb{Z}$, then $\varphi^{-1}([-M, M])$ is the union of finitely many translates of $\ker(\varphi)$ and the latter is a subgroup of infinite index in $G$. If $G$ is moreover free, then the results of "Cogrowth of groups and simple random walks" by Wolfgang Woess apply, so that the considered annular density 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
    Commented Oct 4, 2018 at 21:11
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    $\begingroup$ Thanks, the reference to Woess' work is useful. It resolves one of the key examples that I had in mind: the projection to one of the coordinates after abelianising a free group. As for your suggested map on $\mathbb{Z}^2$ - I think it's clear that the result holds in this case. Intuitively, the preimage of a bounded interval intersected the $n$ ball in $\mathbb{Z}^2$ looks like a finite strip. Using this discription of this preimage, it is easy to show the result. The difficulty seems to be in describing this preimage for an arbitrary hyp-group. $\endgroup$
    – Zestylemonzi
    Commented Oct 5, 2018 at 9:29
  • $\begingroup$ In "Relative growth series in some hyperbolic groups", Richard Sharpe shows that a subsequence of the ratio of interest tends to zero in the case $\varphi(G) \simeq \mathbb{Z}$ and $G$ belongs to some family of hyperbolic groups including hyperbolic surface groups. In "Local limit theorems for free groups", he scrutinizes the asymptotic behavior of ratios which are relevant to the case where $G$ is a free group and $\varphi(G)$ is dense in $\mathbb{R}$. $\endgroup$
    – Luc Guyot
    Commented Oct 7, 2018 at 20:44
  • $\begingroup$ Many thanks again - another great reference. In fact, after some thought, it looks like, that for my purposes, it suffices to show that the liminf of the quotient converges to 0. Hence the subsequence condition proved in Sharp's work would be enough. I'll try to generalise this result to arbitrary hyp groups. Cheers! $\endgroup$
    – Zestylemonzi
    Commented Oct 8, 2018 at 9:18

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