The tameness conjecture for omega-categorical groups and rings

An omega-categorical structure is a structure whose theory has a unique countable model up to isomorphism. A group or ring is tame or supertame in the sense used in the paper's model-theoretic framework. The tameness conjecture. (i) A tame omega-categorical group or ring is virtually nilpotent. (ii) A supertame omega-categorical group is virtually finite-by-abelian, and a supertame omega-categorical ring is virtually finite-by-null. These statements propose structural classifications of omega-categorical groups and rings under increasingly strong model-theoretic tameness assumptions; the source presents them as a meta-conjecture, and no resolution is supplied here.