79 research outputs found
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 -stable, -categorical structures and of the random
graph. In this connection, we also show that the infinite symmetric group
has a unique non-trivial separable group topology. For several
interesting groups we also establish Serre's properties (FH) and (FA)
Selective covering properties of product spaces
We study the preservation of selective covering properties, including classic
ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others,
under products with some major families of concentrated sets of reals.
Our methods include the projection method introduced by the authors in an
earlier work, as well as several new methods. Some special consequences of our
main results are (definitions provided in the paper): \be
\item Every product of a concentrated space with a Hurewicz \sone(\Ga,\Op)
space satisfies \sone(\Ga,\Op). On the other hand, assuming \CH{}, for each
Sierpi\'nski set there is a Luzin set such that L\x S can be mapped
onto the real line by a Borel function.
\item Assuming Semifilter Trichotomy, every concentrated space is
productively Menger and productively Rothberger.
\item Every scale set is productively Hurewicz, productively Menger,
productively Scheepers, and productively Gerlits--Nagy.
\item Assuming \fd=\aleph_1, every productively Lindel\"of space is
productively Hurewicz, productively Menger, and productively Scheepers. \ee
A notorious open problem asks whether the additivity of Rothberger's property
may be strictly greater than \add(\cN), the additivity of the ideal of
Lebesgue-null sets of reals. We obtain a positive answer, modulo the
consistency of Semifilter Trichotomy with \add(\cN)<\cov(\cM).
Our results improve upon and unify a number of results, established earlier
by many authors.Comment: Submitted for publicatio
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