Eventual non-isolation of generic types of simple superstable groups and the $\omega^\omega$ rank threshold for countable models

For a countable complete superstable theory $T$ , Tanović (2013) proved that if $U(T)\ge\omega^\omega$ and the generic type of every infinite simple group definable in $T^{eq}$ of $U$ -rank below $\omega^\omega$ is eventually strongly non-isolated (ESN), then $T$ has $2^{\aleph_0}$ countable models. The paper asks whether the generic type of every simple superstable group is at least eventually non-isolated (its Question 1) and conjectures the strong ESN property for generics of simple superstable groups of $U$ -rank $\omega^{\alpha+1}\cdot n$ (its Conjecture 2). Both remain open: the August 2025 survey by Pillay and Tanović re-poses Question 1 verbatim as an open problem, and the unconditional rank-threshold conjecture $U(T)\ge\omega^\omega\Rightarrow I(T,\aleph_0)=2^{\aleph_0}$ (Tanović 2007) is still proved only for trivial and one-based theories. This record asks for a determination of the isolation status of generics of simple superstable groups; an affirmative answer to the strong form below $\omega^\omega$ would make the threshold theorem unconditional via Theorem 1, yielding the statement that every superstable theory with few countable models has Lascar rank strictly below $\omega^\omega$ .

Background

For a complete first-order theory $T$ in a countable language, $I(T,\aleph_0)$ denotes the number of countable models of $T$ up to isomorphism; it is always at most $2^{\aleph_0}$ . Vaught's conjecture predicts that $I(T,\aleph_0)$ is either countable or $2^{\aleph_0}$ , and theories with $I(T,\aleph_0)<2^{\aleph_0}$ are said to have few countable models. The conjecture is confirmed for $\omega$ -stable theories (Shelah–Harrington–Makkai 1984) and for superstable theories of finite $U$ -rank (Buechler 2008), but remains open for superstable theories in general.

A theory is stable if over every parameter set there are only as many complete types as the cardinality of that set (informally: the theory has few types), and superstable if a stronger iteration of this condition holds; superstability sits one step below $\omega$ -stability in the stability hierarchy. In a superstable theory, complete types carry the Lascar rank $U(p)$ , an ordinal-valued rank measuring the depth of forking — forking being the stability-theoretic notion of dependence between a type and a parameter set. $U(T)$ is the supremum of $U(p)$ over all complete types; ordinals are compared via Cantor normal form, and $\omega^\omega$ is the first limit of the monomials $\omega^n$ . Throughout one works in $T^{eq}$ , the canonical expansion of $T$ by imaginary sorts (quotients by definable equivalence relations), so that sets and groups interpretable in $T$ become definable.

A stable group is a group definable, possibly with parameters, in a stable theory. A generic type of a stable group $G$ is a complete type over the group whose finitely many left translates cover $G$ ; in a connected stable group the generic is unique up to translation, and the $U$ -rank of $G$ is by definition the $U$ -rank of its generic type. A simple group is a group with no nontrivial proper normal subgroups; the groups considered here are infinite simple groups definable in $T^{eq}$ .

A type $p\in S(A)$ (a complete type over the finite or small parameter set $A$ ) is isolated if a single formula generates it. The following isolation notions drive the countable-models theory: $p$ is eventually non-isolated (eni) if some nonforking extension of $p$ over a finite set $B\supseteq A$ is non-isolated; $p$ is strongly non-isolated if, for every finite $B$ and every $n$ , the types over $AB$ almost orthogonal to $p$ are dense in $S_n(AB)$ (almost orthogonal meaning that any realization of one is forking-independent of any realization of the other over $AB$ ); and $p$ is eventually strongly non-isolated (ESN) if some nonforking extension of $p$ over a finite set is strongly non-isolated. ESN implies eni, and the two notions coincide for types of $U$ -rank 1. A strongly regular type — a non-algebraic type whose distinct realizations are pairwise dependent, a higher-rank analogue of a strongly minimal type — that is not eni is called NENI. Tanović (2007) proved a dichotomy: a type is ESN if and only if it is orthogonal to every NENI type whose domain is a finite extension of its own domain; and that the existence of an infinite family of regular ESN types over finite domains with $U$ -ranks $\omega^n$ forces $I(T,\aleph_0)=2^{\aleph_0}$ .

Tanović (2007) conjectured that $U(T)\ge\omega^\omega$ implies $I(T,\aleph_0)=2^{\aleph_0}$ — equivalently, that a superstable theory with few countable models has $U(T)<\omega^\omega$ — and proved it for trivial theories and for one-based theories (one-basedness is a global local-modularity condition; complete theories of modules over a fixed ring are one-based). Low and Pillay (1992) had shown that a superstable, non- $\omega$ -stable theory with few countable models must contain a definable group with locally modular regular generic types; in particular, a superstable theory with no infinite definable groups satisfies Vaught's conjecture. Definable groups are therefore the locus of the remaining difficulty.

The 2013 paper analyzes geometrically how the required ESN families can fail to exist. If they do fail, then — via internally isolated types, their Grassmannian types $p^{(n)}$ (types of names of $n$ -dimensional subspaces of the pregeometry induced on the locus of a regular type $p$ ), and binding groups (type-definable automorphism groups of the locus of a type) — a hypothetical NENI type of $U$ -rank $\omega^{\alpha+1}$ produces a type-definable group acting transitively, indeed 2-transitively, whose $\alpha$ -connected component (a standard connected-component construction for superstable groups) has trivial center; classical stable-group theory then yields a simple, $q$ -connected group or a $q$ -connected field of $U$ -rank about $\omega^\alpha$ . The field case is closed inside the paper: the generic type of a superstable field of $U$ -rank below $\omega^\omega$ is ESN, by a vector-space dimension-counting argument over definable subfields. For simple groups the analogous statement is unknown beyond finite rank: a simple group of finite $U$ -rank is $\aleph_1$ -categorical but not $\aleph_0$ -categorical (a classical fact recalled in the paper), which makes its generic type ENI and, at rank 1, ESN; at infinite rank no structure theory of comparable strength is known. Theorem 1 of the paper states: if $T$ is superstable, $U(T)\ge\omega^\omega$ , and the generic type of every simple group definable in $T^{eq}$ of $U$ -rank smaller than $\omega^\omega$ is ESN, then $I(T,\aleph_0)=2^{\aleph_0}$ . The paper asks its Question 1 (is every such generic at least eventually non-isolated?) and poses Conjecture 2 (the generic of a simple superstable group of $U$ -rank $\omega^{\alpha+1}\cdot n$ is ESN). The question was re-posed verbatim as an open problem (Problem 1) in the August 2025 survey by Pillay and Tanović, which notes that the $U$ -rank $<\omega^\omega$ bound for theories with few countable models would follow without any one-basedness assumption given an affirmative answer.

Problem Statement

Let $T$ be a complete, superstable theory in a countable language, having infinite models, and work in $T^{eq}$ . The problem, posed in the source paper, has a core question and a strong refinement. The two sub-questions have different domains — (1) concerns every infinite simple group definable in $T^{eq}$ , whereas (2) is restricted to groups of monomial $U$ -rank $\omega^{\alpha+1}\cdot n$ — so neither sub-question implies the other. An affirmative answer to the strong form for groups of $U$ -rank below $\omega^\omega$ supplies exactly the missing hypothesis of the paper's Theorem 1:

(1) Question 1 (weak form). Is the generic type of every infinite simple group definable in $T^{eq}$ eventually non-isolated — that is, does some nonforking extension of the generic type over a finite parameter set fail to be isolated?

(2) Conjecture 2 (strong form). Is the generic type of every simple superstable group of $U$ -rank $\omega^{\alpha+1}\cdot n$ (a single monomial term of Cantor normal form) eventually strongly non-isolated (ESN)?

An affirmative answer to (2) for groups of $U$ -rank below $\omega^\omega$ supplies exactly the instances of the hypothesis of Theorem 1 of the source paper that its proof consumes — the proof applies the hypothesis only to the simple, $q$ -connected groups of monomial $U$ -rank $\omega^{\alpha+1}\cdot m$ produced by its Proposition 3.4, the finite-rank instances being the case the paper itself recalls as classically settled — and would thereby immediately yield the unconditional rank-threshold statement: every countable complete superstable theory with $U(T)\ge\omega^\omega$ has $I(T,\aleph_0)=2^{\aleph_0}$ — equivalently, every superstable theory with fewer than $2^{\aleph_0}$ countable models has $U(T)<\omega^\omega$ (the conjecture from Tanović's 2007 paper, currently established only for trivial and one-based theories).

Determine the truth of (1) and (2). An affirmative answer to (2) for groups of $U$ -rank below $\omega^\omega$ would make the threshold theorem unconditional via Theorem 1.

References

Predrag Tanović, "Simple groups and the number of countable models", Archive for Mathematical Logic 52 (2013) 779–791. DOI: 10.1007/s00153-013-0343-x. https://doi.org/10.1007/s00153-013-0343-x (preprint arXiv:1304.7099, https://arxiv.org/abs/1304.7099)
Predrag Tanović, "Non-isolated types in stable theories", Annals of Pure and Applied Logic 145 (2007) 1–15. DOI: 10.1016/j.apal.2006.05.014. https://doi.org/10.1016/j.apal.2006.05.014
Anand Pillay and Predrag Tanović, "The number of countable models of first-order theories" (2025), expository survey submitted to the Journal of Open Mathematical Problems. arXiv:2508.06854. https://arxiv.org/abs/2508.06854
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Dejan Ilić, Slavko Moconja and Predrag Tanović, "Groups with finitely many countable models", Publications de l'Institut Mathématique 97 (2015) 33–41. DOI: 10.2298/PIM140318001I. https://doi.org/10.2298/PIM140318001I