Generic automorphisms and extension lemmas

Generic automorphisms are a common theme when one works with auto-morphism groups of countable structures. They play a pivotal role in reconstruction results. In these notes we shall define and explain a notion of generic, due to John Truss, which is highly relevant to reconstruction. The idea is that a generic automorphism acts on the elements of the structure in the least special way, that is, it should exhibit any finite behaviour that is consistent in the structure. Throughout the talk, we shall deal with automorphisms of countable structures. So let M be countable, and let G = Aut(M) be its automorphism group. There is a standard topology on Aut(M), the one that comes from pointwise convergence. A basis of open sets are the stabilizers of finite sets, that is, subgroups of the form G a := {g ∈ Aut(M) : a g = a} where a is a finite tuple in M, and their cosets. This topology makes M into a topological group-in fact, into a Polish group, that is, a completely metrizable separable topological group. Let us translate some topological notion in terms of behaviour of automor-phisms: • a basic open subset of Aut(M) has the form [p] := {g ∈ Aut(M) : [p] ⊆ g} that is, it will contain all the extensions of a given finite partial iso-morphism p of M;