Intransitive geometries

TL;DR

This paper generalizes Tits' lemma to intransitive geometries, enabling new applications and analyzing amalgams related to intransitive actions of finite orthogonal groups.

Contribution

It extends the connection between simple connectivity and universal completions to intransitive geometries, broadening the scope of Tits' lemma.

Findings

Generalization of Tits' lemma to intransitive geometries

Analysis of amalgams related to intransitive actions of finite orthogonal groups

Potential for numerous applications in geometry and group theory

Abstract

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.