Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

TL;DR

This paper explores the structure of automorphism groups of countable algebraically closed graphs and their relation to endomorphisms of the universal graph, revealing uncountably many maximal subgroups isomorphic to these automorphism groups.

Contribution

It establishes a connection between algebraically closed graphs and endomorphisms of the universal graph, showing uncountably many maximal subgroups isomorphic to automorphism groups.

Findings

Uncountably many maximal subgroups of End R are isomorphic to automorphism groups of countable graphs.

Aut Γ appears as a Schützenberger group in uncountably many ways.

Results extend to universal directed and bipartite graphs.

Abstract

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\Gamma$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $\Gamma$. Further structural information about End $R$ is established including that Aut $\Gamma$ arises in uncountably many ways as a Sch\"{u}tzenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.