A Quasi Curtis-Tits-Phan theorem for the symplectic group

TL;DR

This paper presents a new approach to characterize symplectic groups as universal completions of specific subgroup amalgams, extending classical theorems and providing insights into their geometric and algebraic structure.

Contribution

It establishes a quasi Curtis-Tits-Phan theorem for symplectic groups using amalgamations of low rank subgroups and analyzes their geometric properties.

Findings

Symplectic groups are the universal completion of certain subgroup amalgams.

The associated geometry is simply connected in most cases.

A detailed description of the main exceptional residue is provided.

Abstract

We obtain the symplectic group $\SP(V)$ as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let $\SP(V)$ act flag-transitively on the geometry of maximal rank subspaces of $V$. We show that this geometry and its rank $\ge 3$ residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups.