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

Igor Dolinka, Robert Duncan Gray, Jillian Dawn McPhee, James David Mitchell, Martyn Quick

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
13 Downloads (Pure)

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 Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

Original languageEnglish
Pages (from-to)437-462
Number of pages26
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume160
Issue number3
Early online date21 Jan 2016
DOIs
Publication statusPublished - May 2016

Keywords

  • Existentially closed graphs
  • Algebraically closed graphs
  • Random graph
  • Endomorphism monoid
  • Countable universal graph
  • Countable universal bipartite graph

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