4. Amenability+
Questions pertaining to extensions, relexations, and generalizations of the notions of amenability and related properties (such as the Liouville property). Amenability in conjuction with other properties (is there an amenable ______ group?).
Definition: A group $G$ is supramenable if for every triplet $(G,X,E)$ (where $E$ is a nonempty subset of $X$ and $G \def\actson{\curvearrowright} X$), there is an invariant measure normalized on $E$.
Fact: Subexponential growth implies supramenability.Problem 4.05.
 Q: (Rosenblatt 74’) Is there a supramenable group of exponential growth?
 Is the direct product of $2$ supramenable groups also supramenable?

Definition: A hereditary justinfinite group $G$ is a residually finite group such that every finite index subgroup of it is justinfinite.
Problem 4.1.
Is there an amenable hereditary justinfinite group that is not elementary amenable? Is there a finitely presented one? 
Problem 4.15.
Is there a f.p. amenable infinite simple group? A construction of Juschenko and Monod (2012) gives a finitely generated infinite simple amenable group. Conjecture: The derived subgroup of the full group of the Penrose tiling is a candidate. 
Problem 4.2.
Is there an infinite finitely generated amenable simple nonLiouville group (with respect to symmetric, finitely supported, generating measures)? 
Definition: SQclosure = closure under subgroups and quotients.
Problem 4.25.
 What are "good" SQclosed families of amenable groups?
 What is the SQclosure of the set of bounded automata groups?

Problem 4.3.
Are group actions with polynomialgrowth Schreier graphs extensively amenable?
Extensive amenability is known for recurrent actions. 
Definition: let AE be the smallest class of locally compact groups s..t. the following holds:
 AE contains all compact groups and AE contains all discrete amenable groups;
 closed under the (top) elementray ops.
Problem 4.35.
 is every ever locally compact amenable group in AE?
 Find f.g. amenable groups with infinite commensurated subgroups with trivial normal core.
 Is every commensurated subroup of the Basilica group commensurate with a normal subgroup?

Problem 4.4.
Is there a nonamenable group that has a Liouville action with amenable stabilizers? 
Problem 4.45.
Is there a transitive action $G \def\actson{\curvearrowright} X$ that is $\mu$Liouville, but the action on some orbit of $G\def\actson{\curvearrowright} \subset X \times X$ is not $\mu^2$Liouville? 
Problem 4.5.
Is there a transitive action $G \def\actson{\curvearrowright} X$ that is $\mu$Liouville, but the action on some orbit of $G\def\actson{\curvearrowright} \subset X \times X$ is not $\mu^2$Liouville? 
Problem 4.6.
Is the Liouville property stable under the choice of of symmetric, finitely supported, generating measure? 
Problem 4.75.
Is there an amenable group whose spectrum is a Cantor set?
Remark (Dudko): For Grigorchuk’s group $G$, the spectrum is a union of two intervals: $[1,0]\cup[1/2,1]$; the question is whether there can be something a tad crazier.
Cite this as: AimPL: Amenability of discrete groups, available at http://aimpl.org/amenablediscrete.