Intransitive geometries and fused amalgams

TL;DR

This paper introduces fused amalgams to extend geometric covering theory, enabling the analysis of intransitive geometries related to the group G_2(K) over perfect fields of characteristic 2, resulting in new group amalgamation insights.

Contribution

It develops the concept of fused amalgams to handle intransitive geometries, expanding the applicability of geometric covering theory to new cases.

Findings

Extended geometric covering theory using fused amalgams.

Derived new amalgamation results for G_2(K).

Analyzed geometries with intransitive group actions.

Abstract

We study geometries that arise from the natural $G_2(K)$ action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory by the first and the last author is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group $G_2(K)$.