What is the relation between the Tarski number and growth of a group?

Question

When I was studying the structure of the Grigorchuk group, a question came to my mind and I just had the following information:

We know that every finitely generated group of subexponential growth is amenable so its Tarski number is infinity. Also if a group $G$ contains free group on two generators then we know from Jonsson-Dekker theorm that its Tarski number is 4. And from another theorem we can conclude that $G$ has exponential growth.

But I couldn't understand that, is there any relation between the Tarski number of a group and its growth, or not?

If I want to say what I mean, Let a group $G$ have Tarski number $k$, Is the growth function $\gamma_G (n)$ dependent to $k$ or not? Also if the growth function $\gamma_G (n)=\lambda$, is the Tarski number $\tau (G)$ dependent on $\lambda$ or not?

Answer 1

All exponential growth rates are equivalent but there or groups which have infinite Tarski number (amenable). Any infinite, finitely generated, and non-nilpotent solvable groups is amenable and has exponential growth.

It is also known that there are groups with arbitrarily large Tarski number, initially proved by Mark Sapir in this Math Overflow answer.

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Here is an example if you want to look at "optimal", for some interpretation of optimal, growth functions.

It is not hard to see that $(2n-1)^k$ bounds $F_n$'s $k$-balls volume from below when taking the standard generating set, but $ a c^k$, for $c>2n-1$, will eventually be larger than $k$-balls. You can see a short proof in Pierre de la Harpe's paper Uniform growth in groups of exponential growth example 2.1 that there is not a generating sets which will give you a smaller "optimal base". As you mention, $F_n$ has Tarski number 4 but they have "different growth rates" as you vary $n$ (without the standard equivalent relation).