Fields and Fusions: Hrushovski constructions and their definable groups

An overview is given of the various expansions of fields and fusions of strongly minimal sets obtained by means of Hrushovski's amalgamation method, as well as a characterization of the groups definable in these structures

Generic Automorphisms and Green Fields

We show that the generic automorphism is axiomatisable in the green field of Poizat (once Morleyised) as well as in the bad fields which are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain "bad pseudofinite fields" in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatisation. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories having the definable multiplicity property. We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap in the construction of the bad fiel

Connected components of definable groups and o-minimality I

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin, Newelski, Petrykovski. The examples also give new non G-compact first order theories.Comment: 26 pages. This paper corrects the paper "Groups definable in o-minimal structures: structure theorem, G^000, definable amenability, and bounded orbits" by the first author which was posted in December (1012.4540v1) and later withdraw

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

Forcing a countable structure to belong to the ground model

Suppose that $P$ is a forcing notion, $L$ is a language (in $V$), $\dot{\tau}$ a $P$-name such that $P\Vdash$ "$\dot{\tau}$ is a countable $L$-structure". In the product $P\times P$, there are names $\dot{\tau_{1}},\dot{\tau_{2}}$ such that for any generic filter $G=G_{1}\times G_{2}$ over $P\times P$, $\dot{\tau}_{1}[G]=\dot{\tau}[G_{1}]$ and $\dot{\tau}_{2}[G]=\dot{\tau}[G_{2}]$. Zapletal asked whether or not $P \times P \Vdash \dot{\tau}_{1}\cong\dot{\tau}_{2}$ implies that there is some $M\in V$ such that $P \Vdash \dot{\tau}\cong\check{M}$. We answer this negatively and discuss related issues