On the automorphism group of homogeneous structures

Abstract

A relational structure ${\cal A}$ with a countable universe is

defined to be homogeneous iff every finite partial isomorphism of

${\cal A}$ can be extended to an automorphism of ${\cal A}$.

Endow the universe of ${\cal A}$ with the discrete topology. Then

the automorphism group $Aut({\cal A})$ of ${\cal A}$ becomes a

topological group (with the subspace topology inherited from the suitable topological power of the discrete topology on ${\cal A}$). Recall, that a tuple $\langle g_{0},...,g_{n-1} \rangle$ of elements of $Aut({\cal A})$ is defined to be weakly generic iff

its diagonal conjugacy class (in the group theoretic sense) is

dense in the topological sense, and further, the $\langle g_{0},...,g_{n-1}\rangle$-orbit of each $a \in A$ is finite. \\

Investigations about weakly generic automorphisms have model

theoretic origins (and reasons); however, the existence of weakly

generic automorphisms is closely related to interesting results

in finite combinatorics, as well. \\

\indent In this work we survey some connections between the

existence of weakly generic automorphisms, and finite

combinatorics, group theory and topology. We will recall some

classical results as well as some more recently obtained ones.