Connected components of definable groups and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimality I
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is …
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.
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" …
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.
We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence …
We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an …
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.
Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M . We prove that the quotient $G/{\cal …
Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M . We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K , which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$ . It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$ ), and homotopy equivalent to K . This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is …
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.
In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups deflnable in o-minimal structures, and c) present a structure …
In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups deflnable in o-minimal structures, and c) present a structure theorem for the special case of semi-linear groups, exemplifying their relation with real Lie groups.
Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove …
Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal'cev theorem and we describe the N-definable G-modules and the N-definable rings.
In this paper we develop the theory of covers for locally definable groups in o-minimal structures.
In this paper we develop the theory of covers for locally definable groups in o-minimal structures.
We develop in this paper the theory of covers for Hausdorff properly $\bigvee $-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an …
We develop in this paper the theory of covers for Hausdorff properly $\bigvee $-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an $\N$-definably connected $\N$-definable group $G$ we have $1\to π_1(G)\to \tilde{G}\stackrel{p}\to G\to 1$ in the category of strictly properly $\bigvee $-definable groups with strictly properly $\bigvee $-definable homomorphisms, where $π_1(G)$ is the o-minimal fundamental group of $G$.
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in …
Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in M n that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some M k with the topology induced by M k . Part of this result holds for the wider class of so-called invariant groups: each invariant group G in M n has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as M n .
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and …
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.
Abstract An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We …
Abstract An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.
An argument of A.Borel shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for …
An argument of A.Borel shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of …
Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups. So, by a “definable f -generic” or $dfg$ group we mean a definable group in a saturated model with a global f -generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o -minimal groups in the p -adic context. In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f -generic type, etc. In this paper we will show that every definable f -generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o -minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f -generic subgroup of a definable f -generic group has finite index, and every f -generic type of a definable f -generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f -generic types coincide with almost periodic types in the p -adic case.
Abstract For a group G first order definable in a structure M , we continue the study of the “definable topological dynamics” of G (from [ 9 ] for example). …
Abstract For a group G first order definable in a structure M , we continue the study of the “definable topological dynamics” of G (from [ 9 ] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G ; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below. Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G */( G *) 000 M of G , which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G ; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G */( G *) 000 M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in S G ( M ) are definable; (ii) the kernel of the surjective homomorphism from G */( G *) 000 M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th( M ) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable. In the situation when all types in S G ( M ) are definable, one can just work with the definable (instead of externally definable) objects in the above results.
The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at …
The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well as some new examples. The third is to explore notions of definable map, embedding and isomorphism between these and related quotient objects. The motivation for writing this paper is the recent discovery, via definable groups, of new examples of non G-compact first order theories.
anon his 60th birthday. The 1970's witnessed two attempts to organize large areas of mathematics by intro- ducing a structure in which all of that mathematics took place. Both of …
anon his 60th birthday. The 1970's witnessed two attempts to organize large areas of mathematics by intro- ducing a structure in which all of that mathematics took place. Both of these programs explicitly invoke large cardinal hypotheses; advocates of both programs deny that these hypotheses are necessary. There were a few years ago no extant metatheorems justi- fying these conclusions (but see McLarty (25)). By a metatheorem, I mean a way to tell by looking at a particular proposition whether the purported use of large cardinals is necessary. The natural expectation is that this criteria would be syntactic. But the possibility of geometric criteria arise in (20) and the general model theoretic develop- ment suggests other 'contentual criteria' (Section 4.) We have been unable to develop any reasonable such theorem in the model theoretic case. This paper just explores the territory. As we will see, in both cases, the role of replacement as an intermediary between Zermelo Set Theory and large cardinals arises in the analysis. This work was stimulated by some questions raised by Newelski in (27) concerning the 'absoluteness' of the existence of a bounded orbit and the discussions at about the same time concerning the possible use of large cardinals in Wiles' proof of Fermat's last theorem. On the one hand Grothendieck wanted to provide a framework in which to organize large areas of algebraic geometry. For this he invented the notion of a universe, a large enough set to be closed under the usual algebraic operations. He explicitly developed cohomology theory using the existence of (a proper class of) universes 1 . At about the same time, Shelah introduced his version of a 'universal domain', later dubbed a monster model: C is a saturated model ofT with cardinality a strongly inaccessible cardinal. He writes in (30), 'The assumption on does not, in fact, add any axiom of set theory as a hypothesis to our theorems.' This is a rather unclearly stated metatheorem. 'Which theorems?' An acceptable answer in the short run is, 'the theorems in this book'. And, apparently, the statement is true in that sense. We seek here to clarify what theorems are intended and which are not. We give a brief comparison of the algebraic geometry side (in Section 1.2) relying heavily on (23, 25, 20) and the model theoretic issues (in Subsection1.3). Section 2 I thank the John Templeton Foundation for its support through Project #13152, Myriad Aspects of Infinity, hosted during 2009-2011 at the Centre de Recerca Matematica, Bellaterra, Spain. I thank Rutgers University for support of visits where some of this work was done. 1 The issue we address here of the size of the universe is distinct from the organization via sheaves ad- dressed in (19).
We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, …
We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition , we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to . This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to . We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort and a sort for an abelian group.
Abstract We generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded …
Abstract We generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion.
For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where …
For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as G⁎/(G⁎)M00 and the universal definable G-ambit as the type space SG(M). We also point out the existence and uniqueness of "universal minimal definable G-flows", and discuss issues of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal (Bohr) compactification and universal G-ambit in model-theoretic terms, when G is a topological group (although it is essentially well-known).
Abstract Recall that a group G has finitely satisfiable generics ( fsg ) or definable f -generics ( dfg ) if there is a global type p on G and …
Abstract Recall that a group G has finitely satisfiable generics ( fsg ) or definable f -generics ( dfg ) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$ , respectively. We show that any abelian group definable in a p -adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$ , we show that $G^0 = G^{00}$ , and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$ .
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and …
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.
Abstract We consider definable topological dynamics for NIP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of …
Abstract We consider definable topological dynamics for NIP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of the previous results for this setting.
The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least …
The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L (T) of T, have well-defined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 (2001) 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 (2012) 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint (2012), arXiv:1201.5221v1].
We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite …
We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.
Abstract We consider a group G that does not have the independence property and study the definability of certain subgroups of G , using parameters from a fixed elementary extension …
Abstract We consider a group G that does not have the independence property and study the definability of certain subgroups of G , using parameters from a fixed elementary extension G of G . If X is a definable subset of G , its trace on G is called an externally definable subset . If H is a definable subgroup of G , we call its trace on G an external subgroup . We show the following. For any subset A of G and any external subgroup H of G , the centraliser of A , the A -core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S -invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.
Abstract 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 …
Abstract 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 (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mspace /> <mml:mi>top</mml:mi> <mml:mspace /> </mml:mrow> <mml:mn>00</mml:mn> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mspace /> <mml:mi>top</mml:mi> <mml:mspace /> </mml:mrow> <mml:mn>000</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋁</mml:mo> </mml:math> -definable group topologies on a given $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable group G (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 G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mi>cl</mml:mi> <mml:mspace /> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mn>00</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mspace /> <mml:mi>cl</mml:mi> <mml:mspace /> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mn>000</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for any model M . Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G -invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> in a suitable language (where $${{\mathbb {F}}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> is the free group in 2-generators) for which the $$\bigvee $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋁</mml:mo> </mml:math> -definable group $$H:=\langle X \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>X</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> 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 (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).
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 …
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 $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain ``weak Bohr compactification'' introduced in [24]. In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{topo} = G^{000}_{topo}$. 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 $\bigvee$-definable group topologies on a given $\emptyset$-definable group $G$ (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 $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$.
Thirdly, we give an example of a $\emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $\mathbb{F}_2 \times \mathbb{Z}$ in a suitable language (where $\mathbb{F}_2$ is the free group in 2-generators) for which the $\bigvee$-definable group $H:=\langle X \rangle$ 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).
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ …
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.
We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group …
We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group definable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Subscript exp"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>exp</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {R}_{\exp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which any Lie automorphism is definable.
A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role …
A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role of maximal tori is played by maximal 0-subgroups. Along the way we give structural theorems for solvable groups, linear groups, and extensions of definably compact by torsion-free definable groups.
We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, …
We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if $M$ is a countable, $\omega$-categorical structure and $Aut(M)$ is amenable, as a topological group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is compact, Hausdorff (also over any finite set of parameters), that is $T$ is G-compact. An essentially special case is that if $Aut(M)$ is extremely amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory $T$ can be recovered from its category $Mod(T)$ of models. On the side of definable groups, we prove for example that if $G$ is definable in a model $M$, and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$ and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an earlier paper of the authors.
We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.
We prove (Proposition 2.1) that if μ is a generically stable measure in an NIP (no independence property) theory, and μ(ϕ(x,b))=0 for all b, then for some n, μ(n)(∃y(ϕ(x1,y)∧⋯∧ϕ(xn,y)))=0. As …
We prove (Proposition 2.1) that if μ is a generically stable measure in an NIP (no independence property) theory, and μ(ϕ(x,b))=0 for all b, then for some n, μ(n)(∃y(ϕ(x1,y)∧⋯∧ϕ(xn,y)))=0. As a consequence we show (Proposition 3.2) that if G is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and X is a definable subset of G, then X is generic if and only if every translate of X does not fork over ∅, precisely as in stable groups, answering positively an earlier problem posed by the first two authors.
It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be …
It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group G $G$ is Lie isomorphic to a group definable in an o $o$ -minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group G $G$ has its Levi subgroups with finite center, we prove that in order for G $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452).
We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that …
We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable (and by a result of Simon, this holds unconditionally for $\aleph_0$-categorical theories). We show that if $G$ is definable over $A$ in a hereditarily G-compact theory, then $G^{00}_A=G^{000}_A$.
We also include a brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some (classes of) theories.
Abstract For a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G -flows. In particular, we give a description …
Abstract For a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G -flows. In particular, we give a description of the universal definable G -ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G , that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$ (where G * is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definable G -flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components $G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{*}}000}$ , and show that ${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ is the Δ-definable Bohr compactification of G . We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G . Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.
In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
We initiate a study of definable topological dynamics for groups definable in metastable theories. Specifically, we consider the special linear group $G = SL_2$ with entries from $M = \mathbb{C}((t))$; …
We initiate a study of definable topological dynamics for groups definable in metastable theories. Specifically, we consider the special linear group $G = SL_2$ with entries from $M = \mathbb{C}((t))$; the field of formal Laurent series with complex coefficients. We prove such a group is not definably amenable, find a suitable group decomposition, and describe the minimal flows of the additive and multiplicative groups of $\mathbb{C}((t))$. The main result is an explicit description of the minimal flow and Ellis Group of $(G(M),S_G(M))$ and we observe that this is not isomorphic to $G/G^{00}$, answering a question as to whether metastability is a suitable weakening of a conjecture of Newelski.
This is an updated and slightly expanded version of a tutorial given in the Mini-Course in Model Theory, Torino, February 9-11, 2011. Some parts were previously exposed in the Model …
This is an updated and slightly expanded version of a tutorial given in the Mini-Course in Model Theory, Torino, February 9-11, 2011. Some parts were previously exposed in the Model Theory Seminar of Barcelona. The main goals were to clarify the relation of forking with some versions of splitting in NIP theories and to present the known results on G-compactness, including a full proof of a theorem of E. Hrushovski and A. Pillay on G-compactness of NIP theories over extension bases. The tutorial was given in parallel to a tutorial of H. Adler on forking and dividing over models in NTP2 theories, now written in [2]. Thanks are due to D. Zambella, organizer of the course.
Abstract We introduce and study the model-theoretic notions of absolute connectedness and type - absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups …
Abstract We introduce and study the model-theoretic notions of absolute connectedness and type - absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups ) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification , that is, the class of minimally almost periodic groups .
We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G 00 , etc., are preserved when passing to the theory of the …
We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G 00 , etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M ext , of a model M .In the light of these results we continue the study of the "definable topological dynamics" of groups in NIP theories.In particular we prove the Ellis group conjecture relating the Ellis group to G/G 00 in some new cases, including definably amenable groups in o-minimal structures.
We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these …
We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new classes of examples of groups whose smallest type-definable subgroup of bounded index differs from the smallest invariant subgroup of bounded index. This includes the first known example of a group with this property found by Conversano and Pillay, namely the universal cover of [Formula: see text] (interpreted in a monster model), as well as various examples of different nature, e.g. some central extensions of free groups or of fundamental groups of closed orientable surfaces. As a corollary, we get that both non-abelian free groups and fundamental groups of closed orientable surfaces of genus [Formula: see text], expanded by predicates for all subsets, have this property, too. We also obtain a variant of the example of Conversano and Pillay for [Formula: see text] instead of [Formula: see text], which (as most of our examples) was not accessible by the previously known methods.
Abstract We prove a decomposition of definable groups in o‐minimal structures generalizing the Jordan–Chevalley decomposition of linear algebraic groups. It follows that any definable linear group is a semidirect product …
Abstract We prove a decomposition of definable groups in o‐minimal structures generalizing the Jordan–Chevalley decomposition of linear algebraic groups. It follows that any definable linear group is a semidirect product of its maximal normal definable torsion‐free subgroup and a definable subgroup , unique up to conjugacy, definably isomorphic to a semialgebraic group. Along the way, we establish two other fundamental decompositions of classical groups in arbitrary o‐minimal structures: (1) a Levi decomposition and (2) a key decomposition of disconnected groups, relying on a generalization of Frattini's argument to the o‐minimal setting. In o‐minimal structures, together with ‐groups, 0‐groups play a crucial role. We give a characterization of both classes and show that definable ‐groups are solvable, like finite ‐groups, but they are not necessarily nilpotent. Furthermore, we prove that definable ‐groups ( or prime) are definably generated by torsion elements and, in definably connected groups, 0‐Sylow subgroups coincide with ‐Sylow subgroups for each prime.
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably …
We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic …
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably …
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an …
The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.
In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what …
In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what kinds of properties are admissible in an o-minimal theory?” The motivation of these question begin with the study of simple theories and generalizations of simple theories. In [3] Kim and Pillay prove that the class of simple theories may be described exactly as those theories bearing a notion of independence satisfying various axioms. Thus it is natural to ask, if we weaken the assumptions as to which axioms must hold, what kind of theories do we get? Another source of motivation, also stemming from the study of simple theories, comes from the work of Shelah in [8] and [7]. Here Shelah addresses a “classification” type problem for class of models of a theory, showing that a theory will have the appropriate “structure” type property if one can construct a partially ordered set, satisfying various properties, of models of the theory. Using this criterion Shelah shows that the class of simple theories has this “structure” property, yet also that several non-simple examples do as well (though it should be pointed out that o-minimal theories can not be among these since any theory with the strict order property will have the corresponding “non-structure” property [8]). Thus one is lead to ask, what are the non-simple theories meeting this criterion, and one is once again led to study the types of independence relation a theory might bear. Finally, Shelah in [6] provides some possible definitions of what axioms for a notion of independence one should possibly look for in order to hope that theories bearing such a notion of independence should be amenable closer analysis. In studying all of the above mentioned situations it readily becomes clear that dividing and forking play a central role in all of them, even though we are no longer dealing with the simple case where we know that dividing and forking are very well behaved. All of these considerations lead one to look for classes of non-simple theories of which something is known where one can construct interesting notions of independence and consequently also say something about the nature of forking and dividing in these contexts. Given this one is naturally lead to one of the most well behaved classes of non-simple theories, namely the o-minimal theories.
For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> …
For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak {C}}_{T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every type definable group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating …
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the …
Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.
We study the groups Gal L (T) and Gal KP (T), and the associated equivalence relations E L and E KP , attached to a first order theory T. An …
We study the groups Gal L (T) and Gal KP (T), and the associated equivalence relations E L and E KP , attached to a first order theory T. An example is given where E L ≠ E KP (a non G-compact theory). It is proved that E KP is the composition of E L and the closure of E L . Other examples are given showing this is best possible.
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in …
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2 .
As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. …
As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is reported, C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal [7] clarified for maximally almost periodic groups, that the essential source of the proof of Hilbert's problem for these groups lies in the fact that such groups can be approximated by Lie groups. Here we say that a locally compact group G can be approximated by Lie groups, if G contains a system of normal subgroups {Na} such that G/Na are Lie groups and that the intersection of all Na coincides with the identity e. For the brevity we call such a group a group of type (L) or an (L)-group. In the present paper we shall study the structure of such (L)-groups, and apply the result to solve the Hilbert's problem for a certain class of groups, which contains both compact and solvable groups as special cases. We shall be able to characterize a Lie group G, for which the factor group GIN of G modulo its radical N is compact, completely by its structure as a topological group. The outline of the paper is as follows. In ?1 we study the topological structure of the group of automorphisms of a compact group and prove theorems concerning compact normal subgroups of a connected topological group, which are to be used repeatedly in succeeding sections. In ?2 come some preliminary considerations on solvable groups, whereas finer structural theorems on these groups are, as special cases of (L)-groups, given later. In ?3 we prove some theorems on Lie groups. The theorems here stated are not all new, but we give them here for the sake of completeness, and thereby refine and modify these theorems so as to be applied appropriately in succeeding sections.4 After these preparations we study in ?4 the structure of (L)-groups. In particular, it is shown 'that the study of the local structure and the global topological structure
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is …
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are …
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.