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Asymptotic cones of finitely presented groups
"... Abstract. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that Rrank(S) ≥ 2 and let Γ be a uniform lattice in G. (a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism. (b) If CH fails, then Γ has 2 2ω asymptotic cones up to homeomorphi ..."
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Abstract. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that Rrank(S) ≥ 2 and let Γ be a uniform lattice in G. (a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism. (b) If CH fails, then Γ has 2 2ω asymptotic cones up to homeomorphism. 1.
PseudoFinite Homogeneity and Saturation
 J. SYMBOLIC LOGIC
, 1998
"... When analyzing database query languages a property of theories, the pseudofinite homogeneity property has been introduced and applied (cf. [3]). We show that a stable theory has the pseudofinite homogeneity property just in case its expressive power for finite states is bounded. Moreover, we i ..."
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When analyzing database query languages a property of theories, the pseudofinite homogeneity property has been introduced and applied (cf. [3]). We show that a stable theory has the pseudofinite homogeneity property just in case its expressive power for finite states is bounded. Moreover, we introduce the corresponding pseudofinite saturation property and show that a theory fails to have the finite cover property iff it has the pseudofinite saturation property.
Borel sets with large squares
 Fundamenta Mathematicae
, 1999
"... Abstract. For a cardinal µ we give a sufficient condition ⊕µ (involving ranks measuring existence of independent sets) for: ⊗µ: if a Borel set B ⊆ R × R contains a µsquare (i.e. a set of the form A × A, A  = µ) then it contains a 2 ℵ0square and even a perfect square. And also for ⊗ ′ µ: if ψ ∈ ..."
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Cited by 9 (2 self)
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Abstract. For a cardinal µ we give a sufficient condition ⊕µ (involving ranks measuring existence of independent sets) for: ⊗µ: if a Borel set B ⊆ R × R contains a µsquare (i.e. a set of the form A × A, A  = µ) then it contains a 2 ℵ0square and even a perfect square. And also for ⊗ ′ µ: if ψ ∈ Lω1,ω has a model of cardinality µ then it has a model of cardinality continuum generated in a “nice”, “absolute”way. Assuming MA+2 ℵ0> µ for transparency, those three conditions (⊕µ, ⊗µ and ⊗ ′ µ) are equivalent, and by this we get e.g. ∧ [2 ℵ0
LOCALIZATIONS OF GROUPS
, 2000
"... A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings ..."
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Cited by 7 (2 self)
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A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introduction. For localizations η: A → H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and nonequalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (nonabelian) group. Libman showed that An → SOn−1(R) for a natural embedding of the alternating group An is a localization if n even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [12] that any nonabelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of A’s. The new techniques exploit abelian centralizers of free (nonabelian) subgroups of H which constitute a rigid system of cotorsionfree abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13].
What is a Structure Theory
 Bulletin of the London Mathematical Society
, 1987
"... into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consis ..."
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into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consisted of equalities and inequalities between uncountable cardinals. In my experience most mathematicians find uncountable cardinals depressing, if they have any reaction to them at all. In fact Shelah was quite right to be so happy, but not because of his cardinal inequalities. He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic. That is what I want to discuss here. In brief, Shelah's work is about the notion of a class of structures which has a good structure theory. We all have a rough intuitive notion of what counts as a good structure theory. For example the structure theory of finitely generated abelian groups
Stable Theories With a New Predicate
 J. Symbolic Logic
, 2000
"... Introduction Let M be an Lstructure and A be an infinite subset of M . Two structures can be defined from A: ffl The induced structure on A has a name R' for every ;definable relation '(M) " A n on A. Its language is L ind = fR' j ' = '(x 1 ; : : : ; xn ) an L ..."
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Introduction Let M be an Lstructure and A be an infinite subset of M . Two structures can be defined from A: ffl The induced structure on A has a name R' for every ;definable relation '(M) " A n on A. Its language is L ind = fR' j ' = '(x 1 ; : : : ; xn ) an Lformulag: A with its L ind structure will be denoted by A ind . ffl The pair (M; A) is an L(P )structure, where P is a unary predicate for A and
The selfiterability of L[E
 Journal of Symb. Logic
"... Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with ..."
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Cited by 5 (3 self)
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Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ. As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ> κ> ω1 are cardinals, then ✸ ∗ κ,λ holds true, and if in addition λ is regular, then ✸ + κ,λ holds true. 0 Introduction. If n> 0, then Mn, the least iterable extender model with n Woodin cardinals does not know how to iterate itself (cf. Lemma 1.1). However, if δ1 < · · · < δn are the Woodin cardinals of Mn, then inside Mn, Mnδi+1 is (ω, δi+1)iterable above δi for
Weak forms of elimination of imaginaries
 Math. Logic Quart
"... We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl(A) and clo ..."
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We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl(A) and closed subgroups of the Galois group Aut(acl(A)/A). We also characterize when the topology of the Galois group is the quotient topology. 1