Monotonicity and finite-extension preservation of representation universality

Determine whether representation universality is monotone under passage to subgroups, whether it is preserved by finite extensions, and whether every nonamenable group is representation universal.

Background

The paper uses representation-universal groups to extend comparisons involving the free group, but does not establish general structural principles governing representation universality under standard group-theoretic operations.

The unresolved issues concern restriction to subgroups, passage through finite extensions, and the broad conjectural possibility that nonamenability alone might imply representation universality.