Katrin Tent

On the isometry group of the Urysohn space

We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.

Asymptotic cones and ultrapowers of lie groups

§1. Introduction . Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov. §2. Definitions . The intuitive idea behind Gromovs concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/ n for n ϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromovs original definition, the asymptotic cone existed only for a rather restricted class of spaces.

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon.It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL3(K),PSp4(K) orG2(K) for some algebraically closed fieldK.Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we showTheorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field.Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed.We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.

Moufang polygons and irreducible spherical BN-pairs of rank 2, I

Let G be a group with an irreducible spherical BN-pair of rank 2 satisfying the additional condition: ð * Þ There exists a normal nilpotent subgroup U of B with B ¼ TU; where T ¼ B-N and jW ja16 forthe Weyl gr oup W ¼ N=B-N: We show that G corresponds to a Moufang polygon and hence is essentially known. r 2003 Elsevier Science (USA). All rights reserved.

THE FREE PSEUDOSPACE IS N -AMPLE, BUT NOT ( N + 1)-AMPLE

We give a uniform construction of free pseudospaces of dimension n extending work by Baudisch and Pillay. This yields examples of

The isometry group of the bounded Urysohn space is simple

\omega

Special abelian Moufang sets of finite Morley rank

-stable theories which are n-ample, but not (n+1)-ample. The prime models of these theories are buildings associated to certain right-angled Coxeter groups.

Split BN-Pairs of Rank at Least 2 and the Uniqueness of Splittings

We show that the isometry group of the bounded Urysohn space is simple. This extends previous work by the authors.

Split BN-pairs of rank 2: the octagons

Abstract Moufang sets are split doubly transitive permutation groups, or equivalently, groups with a split BN-pair of rank one. In this paper, we study so-called special Moufang sets with abelian root groups, under the model-theoretic restriction that the groups have finite Morley rank. These groups have a natural base field, and we classify them under the additional assumption that the base field is infinite. The result is that the group is isomorphic to PSL2(K) over some algebraically closed field K.

Split BN-pairs of finite Morley rank

Abstract Let (G, B, N) be a group with an irreducible spherical BN-pair of rank at least 2, and let U be a nilpotent normal subgroup of B such that B = U (B ⋂ N). We show that U is unique with respect to B. As a corollary, we obtain a complete classification of all irreducible spherical split BN-pairs of rank at least two.