Andreas Baudisch

A new uncountably categorical group

We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial.

A free pseudospace

In this paper we construct a non- CM -trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a non CM -trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the original point of the notion of CM -triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non- CM -triviality will come from the behaviour of three orthogonal regular types. A stable theory is said to be CM -trivial if whenever A ⊆ B and acl( Ac ) ∩ acl( B ) = acl( A ) in T eq , then Cb(stp( c / A )) ⊆ Cb(stp( c / B )). ( An infinite stable field will not be CM -trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” were CM -trivial. The notion was studied further in [6] where it was shown that CM -trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8].

Die böse Farbe

We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

Decidability and stability of free nilpotent lie algebras and free nilpotent p-groups of finite exponent

Abstract Our main result is the decidability and ω-stability of free c th nilpotent p -groups of finite exponent ( c p ).

Fusion over a vector space

Let T1 and T2 be two countable strongly minimal theories with the DMP whose common theory is the theory of vector spaces over a fixed finite field. We show that T1 ∪ T2 has a strongly minimal completion.

Stable Actions of Torsion Groups and Stable Soluble Groups

of centralizers of subsets Xi of G has length r at most n. Moreover each definable quotient of G is stable, and so satisfies a chain condition of the above type. In favourable circumstances this information alone is sufficient to give quite precise information about the structure of a stable group. For example, it was proved by Bryant and Hartley [l] that every soluble torsion group satisfying the minimal condition for centralizers is an extension of a nilpotent normal subgroup by an abelian-by-finite group of finite Priifer rank, and it follows a fortiori that stable soluble groups which are torsion groups have this structure. Here we shall show that the same conclusion holds for a somewhat wider class of stable soluble groups. We shall prove this by combining the above result for torsion groups with a result on actions of abelian torsion groups on abelian groups. The examination of two abelian groups together with an action of one of them on the other, all defined in a structure with a stable elementary theory, frequently plays a role in the study of stable groups: one important 453 0021-8693/92

Closures in ℵ 0 -categorical bilinear maps

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A model-theoretic study of right-angled buildings

It is possible to define a combinatorial closure on alternating bilinear maps with few relations similar to that in [2]. For the do-categorical case we show that this closure is part of the algebraic closure. ?

The additive collapse

We study the model theory of countable right-angled buildings with infinite residues. For every Coxeter graph we obtain a complete theory with a natural axiomatisation, which is

Mekler's construction preserves CM-triviality

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