A Tarski Monster for a prime $p$ is usually defined as an infinite simple group whose proper non-trivial subgroups are cyclic of order order $p$.
In "Survey of some results deduced with the help of Ol'shanskii's technique. Part I" by O. Bogopolski one can read the following.
The following definition is more general than that given in Wikipedia.
Definition 2.1.(see [6, Introduction]) A group $G$ is called Tarski Monster if it is infinite, simple and all proper subgroups of $G$ are finite cyclic.
Note that any Tarski Monster is necessarily finitely generated and non-amenable.
For me, this sounds as if the definition of Tarski Monster given in that generality immediately implies non-amenability for such groups.
In "Non-amenable finitely presented torsion-by-cyclic groups" by A.Yu. Ol'shanskii and M.V.Sapir, however, we only have the following statement:
He [Ol'shanskii] proved that the groups with all proper subgroups cyclic constructed by him, both torsion-free and torsion (the so called "Tarski monsters"), are not amenable.
This seems to be a weaker statement as there might exist Tarski monsters which are not obtained by the construction. (A similar statement is also found in Ol'shanskii's book "Geometry of Defining Relations in Groups").
Finally, "On the problem of the existence of an invariant mean of a group" by Ol'shanskii starts with the theorem
Theorem. There exists a non-amenable group $G$, whose proper subgroups are all cyclic.
Of course, there might be research progrees in between these statements, anyway I would like to know:
- Is it true that every Tarski Monster is non-amenable (either given the the usual definition or the the more general one given by Bogopolski)?
- If 1. is affirmative, then how do you prove this statement? I understand that Ol'shanskii used Grigorchuk's cogrowth criterion (amenability in a group $G$ generated by $t$ elements is equivalent to $\sqrt[n]{\gamma_n} \to 2t-1$ where $\gamma_n$ is the number of words of length $\leq n$ which become trivial in $G$) to obtain his results in this direction, but these all seem to rely on his construction method which might not be exhaustive (or is it?).