Turbulence, amalgamation and generic automorphisms of homogeneous structures

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges-Hodkinson-Lascar-Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many $\omega$-stable, $\aleph_0$-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group $S_\infty$ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA)

Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones

In this paper we explore the extent to which the algebraic structure of a monoid $M$ determines the topologies on $M$ that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids. If $M$ is a topological monoid such that every homomorphism from $M$ to a second countable topological monoid $N$ is continuous, then we say that $M$ has \emph{automatic continuity}. We show that many well-known monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid $\mathbb{N}^\mathbb{N}$; the full binary relation monoid $B_{\mathbb{N}}$; the partial transformation monoid $P_{\mathbb{N}}$; the symmetric inverse monoid $I_{\mathbb{N}}$; the monoid Inj$(\mathbb{N})$ consisting of the injective functions on $\mathbb{N}$; and the monoid $C(2^{\mathbb{N}})$ of continuous functions on the Cantor set. We show that the pointwise topology on $\mathbb{N}^\mathbb{N}$, and its analogue on $P_{\mathbb{N}}$, are the unique Polish semigroup topologies on these monoids. The compact-open topology is the unique Polish semigroup topology on $C(2^\mathbb{N})$ and $C([0, 1]^\mathbb{N})$. There are at least 3 Polish semigroup topologies on $I_{\mathbb{N}}$, but a unique Polish inverse semigroup topology. There are no Polish semigroup topologies $B_{\mathbb{N}}$ nor on the partitions monoids. At the other extreme, Inj$(\mathbb{N})$ and the monoid Surj$(\mathbb{N})$ of all surjective functions on $\mathbb{N}$ each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on $\mathbb{N}^\mathbb{N}$, $P_{\mathbb{N}}$, and Inj$(\mathbb{N})$ coincide with the pointwise topology; and we characterise the Zariski topology on $B_{\mathbb{N}}$. In Section 7: clones.Comment: 51 pages (Section 7 about clones was added in version 4

Generic automorphisms and graph coloring

AbstractThe question of whether or not a given countable arithmetically saturated model of Peano Arithmetic has a generic automorphism is shown to be very closely connected to Hedetniemi's well-known conjecture on the chromatic number of products of graphs