50 research outputs found
External definability and groups in NIP theories
We prove that many properties and invariants of definable groups in NIP
theories, such as definable amenability, G/G^{00}, etc., are preserved when
passing to the theory of the Shelah expansion by externally definable sets,
M^{ext}, of a model M. In the light of these results we continue the study of
the "definable topological dynamics" of groups in NIP theories. In particular
we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in
some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were
corrected. Journal of the London Mathematical Society, accepte
Elementary classes of finite VC-dimension
Let U be a monster model and let D be a subset of U. Let (U,D) denote
theexpansion of U with a new predicate for D. Write e(D) for the collection of
all subsets C of U such that (U,C) is elementary equivalent to (U,D). We prove
that if e(D) has finite VC-dimension then D is externally definable (i.e. it is
the trace on U of a set definable in an elementary superstructure of U).Comment: Small correction
A note on generically stable measures and fsg groups
We prove that if \mu is a generically stable stable measure in a first order
theory with NIP and mu(\phi(x,b)) = 0 for all b, then \mu^{(n)}(\exists
y(\phi(x_1,y)\wedge ... \wedge \phi(x_n,y))) = 0. We deduce that if G is an fsg
grooup then a definable subset X of G is generic just if every translate of X
does not fork over \emptyset.Comment: 8 page
Stable embeddedness and NIP
We give sufficient conditions for a predicate P in a complete theory T to be
stably embedded: P with its induced 0-definable structure has "finite rank", P
has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson
and Onshuus in the case where P is o-minimal in T.Comment: 10 page
Invariant types in NIP theories
We study invariant types in NIP theories. Amongst other things: we prove a
definable version of the (p,q)-theorem in theories of small or medium
directionality; we construct a canonical retraction from the space of
M-invariant types to that of M-finitely satisfiable types; we show some
amalgamation results for invariant types and list a number of open questions.Comment: Small changes mad
Definable convolution and idempotent Keisler measures
We initiate a systematic study of the convolution operation on Keisler
measures, generalizing the work of Newelski in the case of types. Adapting
results of Glicksberg, we show that the supports of generically stable (or just
definable, assuming NIP) measures are nice semigroups, and classify idempotent
measures in stable groups as invariant measures on type-definable subgroups. We
establish left-continuity of the convolution map in NIP theories, and use it to
show that the convolution semigroup on finitely satisfiable measures is
isomorphic to a particular Ellis semigroup in this context.Comment: v3. 30 pages; minor corrections and clarifications throughout the
article; accepted to the Israel Journal of Mathematic