27 research outputs found
Model theory of finite and pseudofinite groups
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory
Amenability, connected components, and definable actions
We study amenability of definable groups and topological groups, and prove
various results, briefly described below.
Among our main technical tools, of interest in its own right, is an
elaboration on and strengthening of the Massicot-Wagner version of the
stabilizer theorem, and also some results about measures and measure-like
functions (which we call means and pre-means).
As an application we show that if is an amenable topological group, then
the Bohr compactification of coincides with a certain ``weak Bohr
compactification'' introduced in [24]. In other words, the conclusion says that
certain connected components of coincide: .
We also prove wide generalizations of this result, implying in particular its
extension to a ``definable-topological'' context, confirming the main
conjectures from [24].
We also introduce -definable group topologies on a given
-definable group (including group topologies induced by
type-definable subgroups as well as uniformly definable group topologies), and
prove that the existence of a mean on the lattice of closed, type-definable
subsets of implies (under some assumption) that for any model .
Thirdly, we give an example of a -definable approximate subgroup
in a saturated extension of the group in a
suitable language (where is the free group in 2-generators) for
which the -definable group contains no
type-definable subgroup of bounded index. This refutes a conjecture by Wagner
and shows that the Massicot-Wagner approach to prove that a locally compact
(and in consequence also Lie) ``model'' exists for each approximate subgroup
does not work in general (they proved in [29] that it works for definably
amenable approximate subgroups).Comment: Version 3 contains the material in Sections 2, 3, and 5 of version 1.
Following the advice of editors and referees we have divided version 1 into
two papers, version 3 being the first. The second paper is entitled "On first
order amenability
Theories without the tree property of the second kind
We initiate a systematic study of the class of theories without the tree
property of the second kind - NTP2. Most importantly, we show: the burden is
"sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2
then there is a formula with a single variable witnessing this); NTP2 is
equivalent to the generalized Kim's lemma and to the boundedness of ist-weight;
the dp-rank of a type in an arbitrary theory is witnessed by mutually
indiscernible sequences of realizations of the type, after adding some
parameters - so the dp-rank of a 1-type in any theory is always witnessed by
sequences of singletons; in NTP2 theories, simple types are co-simple,
characterized by the co-independence theorem, and forking between the
realizations of a simple type and arbitrary elements satisfies full symmetry; a
Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite
burden) if and only if the residue field is NTP2 (the residue field and the
value group are strong, of finite burden respectively), so in particular any
ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2
theory preserves NTP2.Comment: 35 pages; v.3: a discussion and a Conjecture 2.7 on the
sub-additivity of burden had been added; Section 3.1 on the SOPn hierarchy
restricted to NTP2 theories had been added; Problem 7.13 had been updated;
numbering of theorems had been changed and some minor typos were fixed;
Annals of Pure and Applied Logic, accepte
Model Theory: groups, geometry, and combinatorics
This conference was about recent interactions of model theory with combinatorics, geometric group theory and the theory of valued fields, and the underlying pure model-theoretic developments. Its aim was to report on recent results in the area, and to foster communication between the different communities
Finding groups in Zariski-like structures
We study quasiminimal classes, i.e. abstract elementary classes (AECs) that arise from a quasiminimal pregeometry structure. For these classes, we develop an independence notion, and in particular, a theory of independence in M^{eq}. We then generalize Hrushovski's Group Configuration Theorem to our setting. In an attempt to generalize Zariski geometries to the context of quasiminimal classes, we give the axiomatization for Zariski-like structures, and as an application of our group configuration theorem, show that groups can be found in them assuming that the pregeometry obtained from the bounded closure operator is non-trivial. Finally, we study the cover of the multiplicative group of an algebraically closed field and show that it provides an example of a Zariski-like structure