81 research outputs found
The dynamical hierarchy for Roelcke precompact Polish groups
We study several distinguished function algebras on a Polish group , under
the assumption that is Roelcke precompact. We do this by means of the
model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate
the dynamics of -categorical metric structures under the action of
their automorphism group. We show that, in this context, every strongly
uniformly continuous function (in particular, every Asplund function) is weakly
almost periodic. We also point out the correspondence between tame functions
and NIP formulas, deducing that the isometry group of the Urysohn sphere is
\Tame\cap\UC-trivial.Comment: 25 page
Topological groups, \mu-types and their stabilizers
We consider an arbitrary topological group definable in a structure
, such that some basis for the topology of consists of sets
definable in .
To each such group we associate a compact -space of partial types
which is the quotient of the usual type
space by the relation of two types being "infinitesimally close to
each other". In the o-minimal setting, if is a definable type then it has a
corresponding definable subgroup , which is the stabilizer of
. This group is nontrivial when is unbounded in the sense of
; in fact it is a torsion-free solvable group.
Along the way, we analyze the general construction of and its
connection to the Samuel compactification of topological groups
Remarks on compactifications of pseudofinite groups
We discuss the Bohr compactification of a pseudofinite group, motivated by a
question of Boris Zilber. Basically referring to results in the literature we
point out (i) the Bohr compactification of an ultraproduct of finite simple
groups is trivial, and (ii) the "definable" Bohr compactification of any
pseudofinite group G, relative to a nonstandard model of set theory in which it
is definable, is commutative-by-profinite.Comment: 9 page
Eberlein oligomorphic groups
We study the Fourier--Stieltjes algebra of Roelcke precompact,
non-archimedean, Polish groups and give a model-theoretic description of the
Hilbert compactification of these groups. We characterize the family of such
groups whose Fourier--Stieltjes algebra is dense in the algebra of weakly
almost periodic functions: those are exactly the automorphism groups of
-stable, -categorical structures. This analysis is then
extended to all semitopological semigroup compactifications of such a
group: is Hilbert-representable if and only if it is an inverse semigroup.
We also show that every factor of the Hilbert compactification is
Hilbert-representable.Comment: 23 page
Tame topology of arithmetic quotients and algebraicity of Hodge loci
We prove that the uniformizing map of any arithmetic quotient, as well as the
period map associated to any pure polarized -variation of Hodge
structure on a smooth complex quasi-projective variety , are
topologically tame. As an easy corollary of these results and of
Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of
is a countable union of algebraic subvarieties of (a
result originally due to Cattani-Deligne-Kaplan).Comment: 23 page
Generalized Bohr compactification and model-theoretic connected components
For a group first order definable in a structure , we continue the
study of the "definable topological dynamics" of . The special case when all
subsets of are definable in the given structure is simply the usual
topological dynamics of the discrete group ; in particular, in this case,
the words "externally definable" and "definable" can be removed in the results
described below.
Here we consider the mutual interactions of three notions or objects: a
certain model-theoretic invariant of , which
appears to be "new" in the classical discrete case and of which we give a
direct description in the paper; the [externally definable] generalized Bohr
compactification of ; [externally definable] strong amenability. Among other
things, we essentially prove: (i) The "new" invariant
lies in between the externally definable generalized Bohr compactification and
the definable Bohr compactification, and these all coincide when is
definably strongly amenable and all types in are definable, (ii) the
kernel of the surjective homomorphism from to the definable
Bohr compactification has naturally the structure of the quotient of a compact
(Hausdorff) group by a dense normal subgroup, and (iii) when is NIP,
then is [externally] definably amenable iff it is externally definably
strongly amenable.
In the situation when all types in are definable, one can just work
with the definable (instead of externally definable) objects in the above
results
Tame topology of arithmetic quotients and algebraicity of Hodge loci
In this paper we prove the following results:
We show that any arithmetic quotient of a homogeneous space admits a
natural real semi-algebraic structure for which its Hecke correspondences are
semi-algebraic. A particularly important example is given by Hodge varieties,
which parametrize pure polarized integral Hodge structures.
We prove that the period map associated to any pure polarized variation
of integral Hodge structures on a smooth complex quasi-projective
variety is definable with respect to an o-minimal structure on the relevant
Hodge variety induced by the above semi-algebraic structure.
As a corollary of and of Peterzil-Starchenko's o-minimal Chow
theorem we recover that the Hodge locus of is a countable
union of algebraic subvarieties of , a result originally due to
Cattani-Deligne-Kaplan. Our approach simplifies the proof of
Cattani-Deligne-Kaplan, as it does not use the full power of the difficult
multivariable -orbit theorem of Cattani-Kaplan-Schmid.Comment: 23 pages, final version. arXiv admin note: substantial text overlap
with arXiv:1803.0938
Stabilizers, Measures and IP-sets
The purpose of this simple note is to provide elementary model-theoretic
proofs to some existing results on sumset phenomena and IP sets, motivated by
Hrushovski's work on the stabilizer theorem
On minimal flows. definably amenable groups, and o-minimality
We study definably amenable groups in NIP theories, and answer a question of
Newelski (and also of Chernikov-Simon), by giving an example in the o-minimal
context where weak generic types do not coincide with almost periodic types,
equivalently where the union of the minimal subflows of suitable type spaces is
not closed. We give other positive results in this o-minimal context.Comment: 23 page
On compactifications and product-free sets
A subset of a group is said to be product-free if it does not contain three
elements satisfying the equation . We give a negative answer to a
question of Babai and S\'os on the existence of large product-free sets by
model theoretic means. This question was originally answered by Gowers.
Furthermore, we give a natural and sufficient model theoretic condition for a
group to have a large product-free subset, as well as a model theoretic account
of a result of Nikolov and Pyber on triple products
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