## 5. Other problems

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

[Girgorchuk]**Conjecture:**A finitely presented group is either virtually nilpotent, or contains a free subsemigroup of order $2$. -
#### 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 5.7.

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.

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