Basic and degenerate pregeometries

TL;DR

This paper investigates the structure of basic and degenerate pregeometries with transitive automorphism groups, providing reductions to simpler cases where the group action is faithful and primitive or quasiprimitive.

Contribution

It characterizes basic pregeometries and shows how their study reduces to cases with faithful, primitive, or quasiprimitive group actions, simplifying classification.

Findings

Basic pairs admit no non-degenerate quotients.

Study reduces to cases with faithful, primitive, or quasiprimitive group actions.

Normal quotients lead to similar reductions.

Abstract

We study pairs $(\Gamma,G)$, where $\Gamma$ is a 'Buekenhout-Tits' pregeometry with all rank 2 truncations connected, and $G\leqslant\mathrm{Aut} \Gamma$ is transitive on the set of elements of each type. The family of such pairs is closed under forming quotients with respect to $G$-invariant type-refining partitions of the element set of $\Gamma$. We identify the 'basic' pairs (those that admit no non-degenerate quotients), and show, by studying quotients and direct decompositions, that the study of basic pregeometries reduces to examining those where the group $G$ is faithful and primitive on the set of elements of each type. We also study the special case of normal quotients, where we take quotients with respect to the orbits of a normal subgroup of $G$. There is a similar reduction for normal-basic pregeometries to those where $G$ is faithful and quasiprimitive on the set of elements of each type.