
## 5. Other problems

Everything that didn’t find a home in other sections.
1. #### Problem 5.1.

[Girgorchuk] Conjecture:A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$.
• #### Problem 5.2.

Is there a finitely presented simple group that is not 2-generated?
• #### Problem 5.3.

Is true that every infinite amenable group contains an infinite abelian subgroup?
• #### Problem 5.4.

$G \curvearrowright X$, $p, x, y \in X$. What can be said on $P(x \in O_\Lambda(p) \operatorname{ and } y \in O_{\Lambda}(p))$? Are these events positively/negatively correlated? (Here $O_\Lambda(p)$ is the inverted orbit of $p$.)
• #### Problem 5.5.

Is there a f.p. infinite torsion group? Conjecture:(Grigorchuk) no.
• #### Problem 5.6.

Is the Tarski number of $G \times G$ equal to the Tarski number of $G$?
• #### Problem 5.7.

Is there a group with Tarski number 7?

If this group is amenable, the Folner function of this group is universal bound.
• #### Problem 5.8.

[Grigorchuk] Conjecture:If the Folner function of a group is sub-exponential, then the group is virtually nilpotent.
• #### Problem 5.9.

Are the Hanoi tower groups $H_n$, $n \geq 4$ amenable?

Cite this as: AimPL: Amenability of discrete groups, available at http://aimpl.org/amenablediscrete.