Generic Automorphisms of Homogeneous Structures

Abstract

Let G be a permutation group on the set Ω. Usually we take |Ω| = ℵ0. We seek a notion of genericity for members of G. The idea is that a member of G should be generic if it is in some sense ‘typical’. We argue that the following is the correct definition. Suppose that G is endowed with a metric so that it becomes a complete metric space. It follows that the Baire category theorem holds, and we may use the notions of ‘meagre’ and ‘comeagre’ sets. An element of G is then said to be generic if it lies in a comeagre conjugacy class. The definition is given relative to the topology we have chosen. But in the main case, where G = Aut U for some countable relational structure U, G becomes a complete metric space on defining d(g, h) = Σ {2−n: xng ≠ xnh or xng −1 ≠ xnh−1} where {xn: n ε ο) is an enumeration of the domain of U, and we always take this metric in this case. In other instances we need to specify the topology explicitly. We study the circumstances under which generic elements do or do not exist. In particular we prove their existence in the following cases: G = Symm Ω, G = Aut Гc, with Гc the random C-coloured graph, and G = Aut (Q, >). Our notion of ‘generic’ is related to one studied by D. Lascar.