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?
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Definition: A hereditary just-infinite group G is a residually finite group such that every finite index subgroup of it is just-infinite.
Problem 4.1.
Is there an amenable hereditary just-infinite 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 non-Liouville group (with respect to symmetric, finitely supported, generating measures)? -
Definition: SQ-closure = closure under subgroups and quotients.
Problem 4.25.
- What are "good" SQ-closed families of amenable groups?
- What is the SQ-closure of the set of bounded automata groups?
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Problem 4.3.
Are group actions with polynomial-growth 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?
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Problem 4.4.
Is there a non-amenable 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.