Classify countable strongly Choquet–Deny groups beyond the finitely generated case

Classify all countable groups that are strongly Choquet–Deny without assuming finite generation, extending the classification that finitely generated strongly Choquet–Deny groups are precisely the virtually abelian groups.

Background

The paper introduces strongly Choquet–Deny groups as a strengthening of Choquet–Deny groups and proves that, for countable groups, this property is equivalent to the existence of equivalent invariant strictly convex norms for uniformly bounded representations on separable Banach spaces. It completely classifies the finitely generated case: such a group is strongly Choquet–Deny if and only if it is virtually abelian.

For countable groups that are not finitely generated, the paper establishes only partial information. Virtually abelian groups are shown to be strongly Choquet–Deny, while groups with an ICC quotient are shown not to be strongly Choquet–Deny. A complete classification of the remaining countable groups is left unresolved.