Primitive pseudo-finite permutation groups of finite SU-rank
Ulla Karhumäki, Nicholas Ramsey
TL;DR
The paper proves that a definably primitive pseudo-finite permutation group $(G,X)$ of finite $SU$-rank satisfies a bound $SU(G) leq ho(SU(X))$ for some function $ ho$. The authors leverage a two-pronged strategy: (i) reduce to primitive groups via definable-primitive versus primitive criteria using supersimple group theory and indecomposability tools, and (ii) apply the Liebeck–Macpherson–Tent classification (via ultraproducts) to limit the possible O'Nan–Scott-type structures to affine, almost simple, simple diagonal, or product-action types, followed by explicit rank bounds in each case. They obtain concrete bounds: if ${ m Rad}(G) eq 1$, $SU(G) le SU(X) + (SU(X)^2+1)SU(X)$; if ${ m Rad}(G)=1$, then $SU(G) le 8 SU(X)^2 + 2 SU(X)$ in the almost simple and simple diagonal cases, while simple diagonal actions satisfy $SU(G) le 2 SU(X)$; a general product-action reduction yields the same asymptotic bound. The paper also treats the rank-1 orbit case $SU(X)=1$ completely, showing $SU(G)\in\\{1,2,3" and identifying the corresponding structures, including realizations via pseudo-finite fields and PSL$_2(F)$-type actions. Overall, the work extends Borovik–Cherlin-type bounds to pseudo-finite groups of finite $SU$-rank and clarifies the role of CFSG-based simple groups in this model-theoretic setting.
Abstract
We study definably primitive pseudo-finite permutation groups of finite $SU$-rank. We show that if $(G,X)$ is such a permutation group, then the rank of $G$ can be bounded in terms of the rank of $X$, providing an analogue of a theorem of Borovik and Cherlin in the setting of definably primitive permutation groups of finite Morley rank.