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Let $G$ be a group with a $S$ a finite subset of $G$ generating it, with $\{e\}\in S$ and $S=S^{-1}$, and let $\gamma_G^S$ be the growth function of $G$ respect to $S$, that is, $\gamma_G^S(l)$ is the number of elements of $G$ which can be expressed as a product of $\leq l$ elements of $S$. Call $J(l)=\gamma_G^S(l)-\gamma_G^S(l-1)$.

If $G$ is infinite, is it true that $J(l+1)\geq J(l)$ for $l\geq1$?

I came up with this question and it seems true, but I haven't found a way to prove it.

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  • $\begingroup$ Not necessarily, think about finite groups. $\endgroup$ Commented Feb 16, 2022 at 21:11
  • $\begingroup$ I state in the question that $G$ is infinite, do you mean that there is some construction using finite groups? $\endgroup$
    – Saúl RM
    Commented Feb 16, 2022 at 21:22
  • $\begingroup$ Oh, somehow I did not notice the assumption that $G$ is infinite. $\endgroup$ Commented Feb 16, 2022 at 22:08
  • $\begingroup$ How much experimentation have you done? I'm genuinely curious about how to write a program to compute $J(l)$ quickly $\endgroup$ Commented Feb 17, 2022 at 15:31
  • $\begingroup$ I have not programmed anything, I just tried to prove that it is true for a while and considered a few very basic examples ($\mathbb{R}^n$ and infinite dihedral group with the usual generators), so it could perfectly be false. I don't know if this can be programmed for general groups $\endgroup$
    – Saúl RM
    Commented Feb 17, 2022 at 16:09

1 Answer 1

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This is false: in the article

R. Grigorchuk and P. De La Harpe, On problems related to growth, entropy, and spectrum in group theory, Journal of Dynamical and Control Systems, Volume 3, Number 1, 51-89,

they provide the counterexample $$G=\langle s,t|s^3=t^3=(st)^3=1\rangle, S=\{e,s,t,s^{-1},t^{-1}\}$$ for which $46=J(11)<J(10)=48$ (the notations you are using are related to those of the article by $\gamma^S_G(l)=\beta(G,S;l)$ and $J(l)=\sigma(l)$). (It was a very interesting question, we have been discussing it with friends since yesterday and kinda agreed that looking at the Cayley graph of the group was the best approach, until one of them found this related question on Mathoverflow which guided us to the article and so to the counterexample).

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  • $\begingroup$ How about the same question, but for sufficiently large $l$? In other words, does $J$ always become eventually non-decreasing? $\endgroup$ Commented Feb 17, 2022 at 19:25
  • $\begingroup$ I don't know, I don't even know if the provided counterexample is "minimal" or if you can actually find another one involving lengths smaller than $10$ and $11$. $\endgroup$
    – Balloon
    Commented Feb 17, 2022 at 20:29

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