Solvability of Rank-Two NIP Groups

Definitions and mathematical statement

Let $G$ be a definable group in the model-theoretic setting intended by the report. NIP means that the ambient theory has no formula with the independence property. Under the standard convention, finitely satisfiable generics (FSG) means that there is a global complete type concentrating on $G$ and a small elementary submodel such that every left translate of that type is finitely satisfiable in the submodel. Hereditary FSG commonly requires this property for each definable subgroup, with the parameter convention to be confirmed.

Assume $G$ has NIP, is hereditarily FSG, and has thorn $U$-rank \[ U^{\mathrm b}(G)=2. \] This is the rank based on thorn-forking, not an automatic substitution of ordinary Lascar $U$-rank or Morley rank. The source does not give its precise rank convention.

The original lists two conjectures:

1. Every such group is solvable-by-finite: it has a solvable normal subgroup of finite index.
2. If in addition $G=G^{00}$, then $G$ is solvable. The source calls this Conjecture 2 the weaker conjecture.

Here $G^{00}$ conventionally denotes the smallest type-definable subgroup of bounded index in the sufficiently saturated ambient structure. Bounded means smaller than its saturation cardinal; this is not the same notation as the definable connected component $G^0$. Solvability means that the derived series terminates at the trivial group after finitely many steps.

The source does not specify the ambient language and theory, parameter sets, saturation conventions, or whether hereditary FSG ranges only over definable subgroups or a broader class. These data affect the intended class and should be recovered. No replacement by pure-group NIP, an o-minimal or finite-Morley-rank hypothesis, rank at most two, or definable connectedness is made. The first conclusion does not impose a uniform derived length or index bound, and the second retains the stronger equality with $G^{00}$.

Short English statement

For hereditarily FSG NIP groups of thorn $U$-rank exactly two, the source asks for solvability up to finite index, and for actual solvability under $G=G^{00}$. The ambient definability and hereditary conventions remain to be fixed.

Sources

• Local UnsolvedMath registry, permanent ID 30000679, code catalogue entry, "Solvability of Rank-Two NIP Groups." Proposed year 2007; proposer unspecified. The original labels the clauses Conjecture 1 and Conjecture 2 (weaker).
• Model Theory and Groups (2007), Oberwolfach Reports, pp. 83--138, DOI 10.4171/owr/2007/02; https://doi.org/10.4171/owr/2007/02. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says official-report content was checked and notation, context, and question form normalized without changing the source claim.
• Literature triage checked 2026-08-21: open. It generically calls the item an unresolved problem/conjecture/direction and reports no authoritative complete resolution from the supplied check, without separate results for the clauses. Created 2026-08-25; updated 2026-09-16. Definitions and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.