Title:

Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures

Let Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group Hclasses, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2[superscript alephnaught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript alephnaught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript alephnaught] regular Dclasses of End(Ω) and when Ω = R,D,B there exist 2[superscript alephnaught] Jclasses of End(Ω). Additionally we show that if Ω = R,D then all regular Dclasses contain 2[superscript alephnaught] group Hclasses. On the other hand, we show that when Ω = B,Q there exist regular Dclasses which contain countably many group Hclasses.
