A characterisation of weakly locally projective amalgams related to $A_{16}$ and the sporadic simple groups $M_{24}$ and $He$

TL;DR

This paper characterizes certain weakly locally projective graphs related to sporadic simple groups, identifying unique and multiple amalgams and their properties, especially focusing on cases connected to $M_{24}$, $He$, and $A_{16}$.

Contribution

It classifies specific weakly locally projective amalgams associated with $M_{24}$, $He$, and $A_{16}$, expanding understanding of their structural properties.

Findings

Exactly two amalgams correspond to the examples from $M_{24}$ and $He$.

For the family $
$, if $n extgreater= 5$, there is a unique unfaithful amalgam.

When $n=4$, there are four amalgams, including two faithful ones related to $M_{24}$ and $He$, and one related to $A_{16}$.

Abstract

A simple undirected graph is weakly $G$-locally projective, for a group of automorphisms $G$, if for each vertex $x$, the stabiliser $G(x)$ induces on the set of vertices adjacent to $x$ a doubly transitive action with socle the projective group $L_{n_x}(q_x)$ for an integer $n_x$ and a prime power $q_x$. It is $G$-locally projective if in addition $G$ is vertex transitive. A theorem of Trofimov reduces the classification of the $G$-locally projective graphs to the case where the distance factors are as in one of the known examples. Although an analogue of Trofimov's result is not yet available for weakly locally projective graphs, we would like to begin a program of characterising some of the remarkable examples. We show that if a graph is weakly locally projective with each $q_x =2$ and $n_x = 2$ or $3$, and if the distance factors are as in the examples arising from the rank 3 tilde geometries of the groups $M_{24}$ and $He$, then up to isomorphism there are exactly two possible amalgams. Moreover, we consider an infinite family of amalgams of type $\mathcal{U}_n$ (where each $q_x=2$ and $n=n_x+1\geq 4$) and prove that if $n\geq 5$ there is a unique amalgam of type $\mathcal{U}_n$ and it is unfaithful, whereas if $n=4$ then there are exactly four amalgams of type $\mathcal{U}_4$, precisely two of which are faithful, namely the ones related to $M_{24}$ and $He$, and one other which has faithful completion $A_{16}$.