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Consider the Coxeter group $G_n$ generated by a finite set $\{s_1, ..., s_n\}$ with respect to the relations $s_1^2=...=s_n^2=1$ and $s_is_j=s_js_i$ for $|i-j| \geq 2$ and denote the word length with respect to the generating set by $l$. The growth series of $G_n$ is defined as $\sum_{g\in G_n} z^{l(g)}$. Denote its radius of convergence by $\rho_n$. Is it then true that $\rho_n \rightarrow 0$ as $n\rightarrow \infty $? Intuitively this seems to be clear but I encounter problems trying to prove this.

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