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.
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).