Moufang sets of finite Morley rank of odd type

We show that for a wide class of groups of finite Morley rank the presence of a split $BN$-pair of Tits rank $1$ forces the group to be of the form $\operatorname{PSL}_2$ and the $BN$-pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian $2$-subgroups and show that all such Moufang sets are standard (and thus associated to $\operatorname{PSL}_2(F)$ for $F$ an algebraically closed field of characteristic not $2$) provided the Hua subgroups are nilpotent. Further, we prove that the same conclusion can be reached whenever the Hua subgroups are $L$-groups and the root groups are not simple

Simple groups of Morley rank 5 are bad

By exploiting the geometry of involutions in $N_\circ^\circ$-groups of finite Morley rank, we show that any simple group of Morley rank $5$ is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. The main result is then used to catalog the nonsoluble connected groups of Morley rank $5$

Involutive automorphisms of N∘∘N_\circ^\circ groups of finite Morley rank

We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup

Rank 3 Bingo

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3

Moufang sets of finite Morley rank

We study proper Moufang sets of finite Morley rank for which either the root groups are abelian or the roots groups have no involutions and the Hua subgroup is nilpotent. We give conditions ensuring that the little projective group of such a Moufang set is isomorphic to PSL2(F) for F an algebraically closed field. In particular, we show that any infinite quasisimple L*-group of finite Morley rank of odd type for which (B;N;U) is a split BN-pair of Tits rank 1 is isomorphic to SL2(F) or PSL2(F) provided that U is abelian. Additionally, we show that same conclusion can reached by replacing the hypothesis that U be abelian with the hypotheses that the intersection of B and N is nilpotent and U is definable and without involutions. As such, we make progress on the open problem of determining the simple groups of finite Morley rank with a split BN-pair of Tits rank 1, a problem tied to the current attempt to classify all simple groups of finite Morley rank