3,981 research outputs found
Imaginaries in separably closed valued fields
We show that separably closed valued fields of finite imperfection degree
(either with lambda-functions or commuting Hasse derivations) eliminate
imaginaries in the geometric language. We then use this classification of
interpretable sets to study stably dominated types in those structures. We show
that separably closed valued fields of finite imperfection degree are
metastable and that the space of stably dominated types is strict
pro-definable
The Lascar groups and the 1st homology groups in model theory
Let be a strong type of an algebraically closed tuple over
B=\acl^{\eq}(B) in any theory . Depending on a ternary relation \indo^*
satisfying some basic axioms (there is at least one such, namely the trivial
independence in ), the first homology group can be introduced,
similarly to \cite{GKK1}. We show that there is a canonical surjective
homomorphism from the Lascar group over to . We also notice that
the map factors naturally via a surjection from the `relativised' Lascar group
of the type (which we define in analogy with the Lascar group of the theory)
onto the homology group, and we give an explicit description of its kernel. Due
to this characterization, it follows that the first homology group of is
independent from the choice of \indo^*, and can be written simply as
. As consequences, in any , we show that
unless is trivial, and we give a criterion for the equality of stp and
Lstp of algebraically closed tuples using the notions of the first homology
group and a relativised Lascar group. We also argue how any abelian connected
compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772
On Roeckle-precompact Polish group which cannot act transitively on a complete metric space
We study when a continuous isometric action of a Polish group on a complete
metric space is, or can be, transitive. Our main results consist of showing
that certain Polish groups, namely and
, such an action can never be transitive (unless the
space acted upon is a singleton). We also point out "circumstantial evidence"
that this pathology could be related to that of Polish groups which are not
closed permutation groups and yet have discrete uniform distance, and give a
general characterisation of continuous isometric action of a Roeckle-precompact
Polish group on a complete metric space is transitive. It follows that the
morphism from a Roeckle-precompact Polish group to its Bohr compactification is
surjective
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
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ON WEAK ELIMINATION OF IMAGINARIES, AN APPENDIX OF CASANOVAS AND FARRE'S PAPER AND BASIC EXAMPLES (Model theoretic aspects of the notion of independence and dimension)
We give an explicit proof of Fact 2.2.2 in [CF]. And we give basic examples related to (full)/weak/geometric elimination of imaginaries
Dimension, matroids, and dense pairs of first-order structures
A structure M is pregeometric if the algebraic closure is a pregeometry in
all M' elementarily equivalent to M. We define a generalisation: structures
with an existential matroid. The main examples are superstable groups of U-rank
a power of omega and d-minimal expansion of fields. Ultraproducts of
pregeometric structures expanding a field, while not pregeometric in general,
do have an unique existential matroid.
Generalising previous results by van den Dries, we define dense elementary
pairs of structures expanding a field and with an existential matroid, and we
show that the corresponding theories have natural completions, whose models
also have a unique existential matroid. We extend the above result to dense
tuples of structures.Comment: Version 2.8. 61 page
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