Algebraicity of proper Moufang sets of finite Morley rank (Borovik-Nesin Question B.17)

A Moufang set is the permutation configuration $(X,\{U_x : x \in X\})$ underlying a split $BN$-pair of Tits rank 1: conjugate root groups $U_x$, each fixing $x$ and acting regularly on the remaining points, generate a 2-transitive 'little projective group'. For Moufang sets interpretable in a structure of finite Morley rank (a model-theoretic notion of dimension generalizing algebraic groups over algebraically closed fields), the expected classification is the rank-one case of the Cherlin-Zilber algebraicity program: Borovik-Nesin Question B.17 conjectures that every infinite proper Moufang set of finite Morley rank is isomorphic to $\mathbb{M}(F)$, the Moufang set of the projective line over an algebraically closed field $F$, with little projective group $\operatorname{PSL}_2(F)$. Wiscons (2014) confirmed the conjecture for infinite hereditarily proper Moufang sets of odd type with nilpotent Hua subgroups, and reduced the case of $L$-group Hua subgroups to one explicit exceptional configuration: root groups that are infinite simple groups of degenerate type, together with Hua subgroups of Prüfer 2-rank at least 2.

Background

Groups of finite Morley rank are groups definable in a structure carrying a model-theoretic dimension theory (Morley rank), the prototype being algebraic groups over algebraically closed fields, where Morley rank agrees with Zariski dimension. The Cherlin-Zilber algebraicity conjecture predicts that every infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field; the even- and mixed-type cases were settled affirmatively by Altınel, Borovik and Cherlin in 2008, while degenerate and odd types remain open.

Every simple algebraic group over an algebraically closed field has a $BN$-pair: subgroups $B$ and $N$ generating the group such that the Weyl group $W = N/(B \cap N)$ is a Coxeter group; its rank is the Tits rank, and the $BN$-pair splits when $B = U \rtimes T$ with $U$ nilpotent and normal in $B$. Kramer, Tent and Van Maldeghem proved in 1999 that a simple group of finite Morley rank with a spherical $BN$-pair of Tits rank at least 3, or with a spherical Moufang $BN$-pair of Tits rank 2, is algebraic, and later work based on the classification of Moufang buildings completes the general rank-2 case. The rank-1 case is the frontier, and it is governed by Moufang sets.

A Moufang set, introduced by Tits, is a pair $(X, \{U_x : x \in X\})$ with $|X| \geq 3$ such that each root group $U_x \leq \operatorname{Sym}(X)$ fixes $x$ and acts regularly on $X \setminus \{x\}$, and the $U_x$ form a single conjugacy class in the group $G = \langle U_x : x \in X\rangle$ they generate, the little projective group. Split $BN$-pairs of Tits rank 1 and Moufang sets determine each other. The basic example is $\mathbb{M}(F)$: take $X = \mathbb{P}^1(F)$ with the root groups the unipotent radicals of the point stabilizers in $\operatorname{PSL}_2(F)$; more generally every quadratic Jordan division algebra $J$ yields a Moufang set $\mathbb{M}(J)$. The Moufang set is proper when $G$ is not sharply 2-transitive on $X$, equivalently when its two-point stabilizers, the Hua subgroups, are nontrivial; non-proper Moufang sets correspond to sharply 2-transitive groups. A Moufang set of finite Morley rank is one for which $X$, the root groups, $G$, and the action are interpretable in a structure of finite Morley rank. The Moufang set is special when the defining permutation $\tau$ swapping two distinguished points commutes with inversion on the nonidentity elements of the root group; by a theorem of Segev, every proper Moufang set with abelian root groups is special, so 'special' is the natural home of the abelian theory.

The finite Moufang sets were classified by Hering, Kantor and Seitz in 1972 through the classification of finite groups with a split $BN$-pair of rank 1. The expected infinite, finite-Morley-rank analogue is Borovik-Nesin Question B.17, restated by Wiscons (2014): every infinite proper Moufang set of finite Morley rank is isomorphic to $\mathbb{M}(F)$ for an algebraically closed field $F$. The abelian case is known: De Medts and Tent (2008) proved that special abelian Moufang sets of finite Morley rank with torsion-free root groups come from algebraically closed fields, Wiscons (2010) settled characteristic 2, and Wiscons (2011) completed the elementary-abelian case under the $L$-group hypothesis described below.

The analysis of groups of finite Morley rank splits by the connected component $S^\circ$ of a Sylow 2-subgroup $S$ (Sylow theory holds in this category): a group is of degenerate type if $S^\circ$ is trivial, of even type if $S^\circ$ is nontrivial, nilpotent and of bounded exponent (a 2-unipotent group), of odd type if $S^\circ$ is nontrivial, divisible and abelian (a 2-torus), and of mixed type otherwise. The Prüfer 2-rank of a group is the maximal $n$ such that it contains a subgroup isomorphic to $(\mathbb{Z}(2^\infty))^n$, the $n$-fold direct product of the Prüfer 2-group, i.e. the rank of a 2-torus. The (regular) 2-rank is a different invariant, the maximal rank of an elementary abelian 2-subgroup, and the Prüfer 2-rank is always at most the 2-rank. For an infinite Moufang set of finite Morley rank the little projective group is connected and 2-transitive, and by a result of Borovik and Cherlin it is then of odd or of even type, so 'odd type' excludes only the even-type case. A group of finite Morley rank is an $L$-group if every infinite definable simple section of odd type is isomorphic to an algebraic group over an algebraically closed field (a section being a quotient of a definable subgroup by a definable normal subgroup); the $L$-hypothesis is the standard inductive assumption that lets the odd-type program proceed without a classification of degenerate-type simple groups.

Two further standing conjectures frame the problem. First, De Medts, Segev and Tent conjectured that the root groups of any Moufang set are nilpotent; this is known for Moufang buildings of rank at least 2 (Tent 2004) and in the finite case, but is open for rank-1 Moufang sets, even in finite Morley rank. Second, a root subgroup $V$ of a root group $U$ (a subgroup invariant under a suitable $\mu$-map) induces a sub-Moufang set; a Moufang set is hereditarily proper if every induced sub-Moufang set is again proper. Segev conjectured (Conjecture 2 of his 2009 paper) that, apart from a few small known examples, the induced Moufang sets of a proper Moufang set remain proper, and this conjectural automatic properness is what Wiscons's theorems assume as 'hereditarily proper'.

Within this framework Wiscons (2014) proved: Theorem A, an infinite hereditarily proper Moufang set of finite Morley rank of odd type with nilpotent Hua subgroups is isomorphic to $\mathbb{M}(F)$ for an algebraically closed field $F$ of characteristic not 2; and Theorem B, the same conclusion holds when the Hua subgroups are $L$-groups, unless both of the following occur: every root group is a simple group of degenerate type, and the Hua subgroups have Prüfer 2-rank at least 2. Theorem B narrows the odd-type, hereditarily proper, $L$-group case to this single exceptional configuration, which the paper does not exclude; it also yields that a simple odd-type $K^*$-group of finite Morley rank with a definable split $BN$-pair of rank 1 whose root group has no involutions is $\operatorname{PSL}_2(F)$ with its standard $BN$-pair. No resolution of the general conjecture, of this exceptional configuration, or of the two standing conjectures above is known in the literature.

Problem Statement

Prove or refute Borovik-Nesin Question B.17 in its rank-one form: every infinite proper Moufang set of finite Morley rank is isomorphic to $\mathbb{M}(F)$ for some algebraically closed field $F$, so that its little projective group is isomorphic to $\operatorname{PSL}_2(F)$ in its natural action on the projective line and its root groups are the images of the additive group of $F$. Wiscons's Theorem B leaves open exactly one odd-type, hereditarily proper, $L$-group configuration: every root group an infinite simple group of degenerate type, and every Hua subgroup of Prüfer 2-rank at least 2. A counterexample, i.e. an infinite proper Moufang set of finite Morley rank not isomorphic to any $\mathbb{M}(F)$, would refute the conjecture; the even-type case, the non-hereditarily-proper case, and the case of Hua subgroups that are neither nilpotent nor $L$-groups also remain open. No hypotheses beyond properness, infiniteness, and finite Morley rank may be assumed.

References

1. Joshua Wiscons, "Moufang sets of finite Morley rank of odd type", Journal of Algebra 402 (2014) 479-498. DOI 10.1016/j.jalgebra.2014.01.001; arXiv:1401.2017. https://doi.org/10.1016/j.jalgebra.2014.01.001
2. Alexandre Borovik and Ali Nesin, "Groups of Finite Morley Rank", Oxford Logic Guides 26, The Clarendon Press, Oxford University Press, New York, 1994 (Question B.17). DOI 10.1093/oso/9780198534457.001.0001; Zbl 0816.20001. https://doi.org/10.1093/oso/9780198534457.001.0001
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