17 research outputs found
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
Definable Topological Dynamics in Metastable Theories
We initiate a study of the definable topological dynamics of groups definable in metastable theories. In stable theories, it is known that the quotient of a group G by its type-definable connected component G00 is isomorphic to the Ellis Group of the flow (G(M),SG(M)); we consider whether these results could be extended to the broader metastable setting. Further, the definable topological dynamics of compactly dominated groups in the o-minimal setting is well understood. We investigate to what extent stable domination is a suitable analogue of compact domination in regards to describing the Ellis Group of metastable definable group
Set Theory
This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject
Combinatorial Properties of Finite Models
We study countable embedding-universal and homomorphism-universal structures
and unify results related to both of these notions. We show that many universal
and ultrahomogeneous structures allow a concise description (called here a
finite presentation). Extending classical work of Rado (for the random graph),
we find a finite presentation for each of the following classes: homogeneous
undirected graphs, homogeneous tournaments and homogeneous partially ordered
sets. We also give a finite presentation of the rational Urysohn metric space
and some homogeneous directed graphs.
We survey well known structures that are finitely presented. We focus on
structures endowed with natural partial orders and prove their universality.
These partial orders include partial orders on sets of words, partial orders
formed by geometric objects, grammars, polynomials and homomorphism orders for
various combinatorial objects.
We give a new combinatorial proof of the existence of embedding-universal
objects for homomorphism-defined classes of structures. This relates countable
embedding-universal structures to homomorphism dualities (finite
homomorphism-universal structures) and Urysohn metric spaces. Our explicit
construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Dichotomies in Constraint Satisfaction: Canonical Functions and Numeric CSPs
Constraint satisfaction problems (CSPs) form a large class of decision problems that con- tains numerous classical problems like the satisfiability problem for propositional formulas and the graph colourability problem. Feder and Vardi [52] gave the following logical for- malization of the class of CSPs: every finite relational structure A, the template, gives rise to the decision problem of determining whether there exists a homomorphism from a finite input structure B to A. In their seminal paper, Feder and Vardi recognised that CSPs had a particular status in the landscape of computational complexity: despite the generality of these problems, it seemed impossible to construct NP-intermediate problems
`a la Ladner [72] within this class. The authors thus conjectured that the class of CSPs satisfies a complexity dichotomy , i.e., that every CSP is solvable in polynomial time or is NP-complete. The Feder-Vardi dichotomy conjecture was the motivation of an intensive line of research over the last two decades. Some of the landmarks of this research are the confirmation of the conjecture for special classes of templates, e.g., for the class of undi- rected graphs [55], for the class of smooth digraphs [5], and for templates with at most three elements [43, 84]. Finally, after being open for 25 years, Bulatov [44] and Zhuk [87] independently proved that the conjecture of Feder and Vardi indeed holds.
The success of the research program on the Feder-Vardi conjecture is based on the con- nection between constraint satisfaction problems and universal algebra. In their seminal paper, Feder and Vardi described polynomial-time algorithms for CSPs whose template satisfies some closure properties. These closure properties are properties of the polymor- phism clone of the template and similar properties were later used to provide tractability or hardness criteria [61, 62]. Shortly thereafter, Bulatov, Jeavons, and Krokhin [46] proved that the complexity of the CSP depends only on the equational properties of the poly- morphism clone of the template. They proved that trivial equational properties imply hardness of the CSP, and conjectured that the CSP is solvable in polynomial time if the polymorphism clone of the template satisfies some nontrivial equation. It is this conjecture that Bulatov and Zhuk finally proved, relying on recent developments in universal algebra. As a by-product of the fact that the delineation between polynomial-time tractability and NP-hardness can be stated algebraically, we also obtain that the meta-problem for finite- domain CSPs is decidable. That is, there exists an algorithm that, given a finite relational structure A as input, decides the complexity of the CSP of A