Existence of bad groups of finite Morley rank in the generalized sense of Corredor, Borovik–Poizat, and Jaligot

After Frécon's proof that bad groups in the sense of Cherlin — nonsolvable groups of Morley rank $3$ with no definable subgroup of Morley rank $2$ — do not exist, so that every simple group of Morley rank $3$ is isomorphic to $\operatorname{PSL}_2(K)$ over an algebraically closed field $K$, the existence question remains open for the more general notions of bad group introduced by Corredor and by Borovik–Poizat and pursued by Jaligot: does there exist a connected nonsolvable group of finite Morley rank all of whose proper definable connected solvable subgroups are nilpotent? In a closely related geometric formulation, does there exist a connected full Frobenius group of finite Morley rank, i.e. a group covered by the conjugates of a proper definable malnormal subgroup (a subgroup meeting each of its other conjugates only in the identity) — a ‘sad group’ in the terminology of Poizat and Wagner? Such groups are the canonical potential counterexamples without involutions to the Cherlin–Zilber algebraicity conjecture, and any simple group of Morley rank $4$ or $5$ would necessarily be one, so the question is the exact remaining obstruction to that conjecture in ranks at most $5$. All known elimination theorems assume an abelian Borel subgroup (a maximal connected solvable definable subgroup): rank $3$ (Frécon), and more generally rank $2n+1$ with abelian Borel of rank $n$ (Poizat–Wagner; Wagner), extended by Clausen and Tent to all degenerate-type Frobenius groups with abelian complement. The cases of non-abelian Borel subgroups and of higher ranks are open, and no construction is known.

Background

Morley rank is the model-theoretic analogue of algebraic dimension: an ordinal-valued rank on the definable sets of a first-order structure, finite for all definable sets in totally transcendental theories (for countable languages, equivalently $\omega$-stable theories, those admitting no infinite binary tree of definable subsets). A group of finite Morley rank is a group whose underlying set and group operation are definable in such a structure; the motivating examples are algebraic groups over algebraic closed fields, where Morley rank agrees with Zariski dimension. A subgroup is definable when its underlying set is definable; a definable group is connected when it has no definable proper subgroup of finite index; an involution is an element of order $2$; a group is nilpotent when its lower central chain reaches the identity in finitely many steps (abelian groups are the one-step case); a subgroup $H$ of $G$ is malnormal when $H \cap H^g = \{1\}$ for every $g \in G \setminus H$, where $H^g$ is a conjugate of $H$.

The Cherlin–Zilber algebraicity conjecture asserts that every infinite simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field; informally, abstract dimension theory is supposed to force algebraic structure. It is the central open problem of the model theory of groups. The Borovik programme, modeled on the classification of finite simple groups, has settled major parts — notably, by Altınel, Borovik, and Cherlin, the case of groups containing an infinite elementary abelian subgroup consisting of involutions — but the conjecture remains open in the degenerate case, where the group contains no involution at all.

Cherlin's 1979 analysis of low ranks reduced the rank-$3$ case to a pathological alternative that he named a bad group: a nonsolvable group of Morley rank $3$ with no definable subgroup of Morley rank $2$. Such a group would be simple modulo its finite center, involution-free, and non-algebraic. Nesin showed that in a rank-$3$ bad group the Borel subgroups (the maximal connected solvable definable subgroups) would be nilpotent, self-normalizing, malnormal, and conjugate, and would cover the group by their conjugates; he also showed that such a group acts on an incidence geometry close to a non-Desarguesian projective plane of Morley rank $2$. Baldwin's construction of an almost strongly minimal (essentially of Morley rank $1$) non-Desarguesian projective plane — a projective plane in which Desargues's theorem fails — shows that this rank-$2$ geometry is itself consistent, so no contradiction follows from the geometry alone. No bad group is existentially closed — every finite system of equations and inequations with a solution in some extension group already has a solution in the group (Jaligot–Ould Houcine) — or linear, i.e. isomorphic to a group of invertible matrices over a field (Mustafin–Poizat), and the existence question stayed fully open until Frécon proved, by a direct combinatorial analysis of the lines and planes definable in such a group, that bad groups of Morley rank $3$ do not exist; consequently every simple group of Morley rank $3$ is isomorphic to $\operatorname{PSL}_2(K)$ — the projective special linear group, that is, the group of $2 \times 2$ matrices of determinant $1$ over $K$ modulo its center — for an algebraically closed field $K$. Corredor and Deloro later gave a streamlined, self-contained exposition of this proof using group actions in place of incidence geometries.

The notions of bad group in the current literature are, however, more general than Cherlin's rank-$3$ one, and these are exactly what Frécon's paper explicitly leaves untouched. Corredor, and independently Borovik and Poizat, called bad any nonsolvable connected group of finite Morley rank all of whose proper connected definable subgroups are nilpotent; Jaligot relaxed this to require only that every connected definable solvable subgroup be nilpotent, and identified the essential configuration as that of a full Frobenius group: a group covered by the conjugates of a proper malnormal subgroup, in analogy with Frobenius groups of finite group theory (semidirect products $N \rtimes H$ whose complement $H$ intersects each of its conjugates outside $N$ trivially). Poizat and Wagner call a group carrying such a covering by conjugates of a definable malnormal subgroup a sad group (French: groupe malsain). In a minimal bad group the Borel subgroups are nilpotent, definable, self-normalizing, malnormal, conjugate, and cover the group, so a minimal bad group gives rise to a special full Frobenius group with nilpotent Borel subgroups; Wagner shows conversely that if no special full Frobenius group exists — special meaning, in Wagner's terminology, that its Borel subgroups are precisely its maximal proper definable subgroups — then no full Frobenius group and no bad group exists. By work growing out of ideas of Borovik, Corredor, Nesin, and Poizat, every bad group is of degenerate type: it has no involutions, every element has a unique square root, and no nontrivial element is conjugate to its inverse. This is why bad groups are the canonical candidate counterexamples of degenerate type to the algebraicity conjecture: a bad group has a simple definable subgroup which is still a full Frobenius group, and a simple involution-free group cannot be algebraic.

The generalized question is not idle. Wiscons proved that a simple group of Morley rank $4$ would be a bad group all of whose proper definable subgroups have rank at most $1$, and Deloro and Wiscons proved that a simple group of Morley rank $5$ would be a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. Thus the algebraicity conjecture in Morley ranks at most $5$ is now precisely the bad-group existence question. On the nonexistence side, Poizat and Wagner, and Wagner separately, re-proved and generalized Frécon's theorem by constructing a ‘Frécon set’ inside the group: there is no bad group, and more generally no simple full Frobenius group, of Morley rank $2n+1$ whose Borel subgroups are abelian of rank $n$; Clausen and Tent extended the obstruction to every connected Frobenius group of finite Morley rank of degenerate type with abelian complement of rank $k$, which must have rank at least $2k+1$, with equality forcing the group to split as a semidirect product. All of these arguments build the Frécon set from the commutator pairing on an abelian subgroup and break down when the Borel subgroup is non-abelian or when the rank of the group strictly exceeds twice the rank of its Borel subgroups. Beyond that boundary nothing is known: no elimination theorem, and no candidate construction. The surrounding geometries — Poizat's symétrons (his axiomatic midpoint-convexity spaces adapted to groups without involutions) and Clausen–Tent's mock hyperbolic reflection spaces (incidence structures whose points are the involutions of a group and whose lines are determined by pairs of them, in analogy with reflections in the hyperbolic plane) — organize the constraints but have so far yielded neither a contradiction nor an example.

Problem Statement

The problem is the existence question that Frécon's ‘Bad groups in the sense of Cherlin’ explicitly leaves open: do bad groups of finite Morley rank exist in the generalized senses of Corredor, Borovik–Poizat, and Jaligot?

Primary question. Does there exist a connected nonsolvable group $G$ of finite Morley rank such that every proper definable connected solvable subgroup of $G$ is nilpotent? (This is the Jaligot–Wagner form of the notion; the original Corredor/Borovik–Poizat form — every proper connected definable subgroup nilpotent — is satisfied by any minimal-rank example, so an answer for either form transfers to the other by passing to a minimal counterexample.)

Related geometric formulation. Does there exist a connected group of finite Morley rank containing a proper definable malnormal subgroup whose conjugates cover the group — a full Frobenius group of finite Morley rank, in Poizat–Wagner's terminology a sad group? In a minimal bad group the Borel subgroups form exactly such a covering, and Wagner proves that nonexistence of special full Frobenius groups (full Frobenius groups whose Borel subgroups are precisely their maximal proper definable subgroups) implies nonexistence of bad groups. The correspondence is one-directional on the existence side, however: Wagner shows that a special full Frobenius group whose Borel subgroups are nilpotent is a bad group, and nilpotence of the Borel subgroups of a hypothetical full Frobenius group is not known to be automatic; a constructed full Frobenius group therefore resolves the primary question only when its Borel subgroups are nilpotent, or it contains a bad group.

The question has two possible answers:

(i) Nonexistence: a proof that no such group exists, in any Morley rank, with no restriction on the Borel subgroups (in particular not assuming them abelian).

(ii) Existence: an explicit construction of such a group $G$, together with proofs that the theory of $G$ has finite Morley rank, that $G$ is connected and nonsolvable, and that every proper definable connected solvable subgroup of $G$ is nilpotent (or, for the geometric formulation, that $G$ contains a proper definable malnormal subgroup whose conjugates cover $G$ and that the Borel subgroups of $G$ are nilpotent, so that $G$ is a bad group in the sense of the primary question).

Scope conventions. ‘Finite Morley rank’ is understood in the standard sense of the literature: $G$ is a group definable in a possibly enriched structure all of whose definable sets have finite Morley rank, and all subgroup conditions refer to definability in that structure. The stakes are explicit: by Wiscons and Deloro–Wiscons, any simple group of Morley rank $4$ or $5$ would be a bad group, so a nonexistence proof settles the Cherlin–Zilber algebraicity conjecture in all Morley ranks at most $5$; and since a bad group has a simple definable full Frobenius subgroup, which is involution-free and hence not algebraic, an existence construction would refute the conjecture.

References

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