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Let $G$ be a group with a finite symmetric set $S$ of generators. Let $\ell_S(x)$ denote the word-length of a given $x\in G$. For $s\in\mathbb C$ set $$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s}, $$ where $G^*=G\smallsetminus\{1\}$. It may happen that this sum converges for some $s$. By Gromov's Theorem on groups of polynomial growth, this is the case if and only if the group has a nilpotent subgroup of finite index. In the very few cases I have computed, this function turned out to be a linear combination of Riemann zetas with shifted arguments.

My question is a reference request: Has this kind of Dirichlet series been investigated? If so, I would like to have references.

Thank you.

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  • $\begingroup$ Minor observation : there can be different groups with the same zeta function. Z^2 and the free group on two generators where $ xy = y^{-1}x$, for instance. $\endgroup$
    – Asvin
    Commented May 17, 2019 at 8:38
  • $\begingroup$ People looked at similar series $\sum_x s^{-\ell(x)}$. $\endgroup$
    – Misha
    Commented May 17, 2019 at 17:17

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Pierre de la Harpe's Topics in Geometric Group Theory has a chapter (VI) on series using the length function on words, but these may be traditional power series. Zeta functions are usually introduced when there is some sort of product formula, as in Euler's product formula for the Riemann zeta-function. So does this happen here, at least for some reasonable class of groups?

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