Definable Compactness and Definable Subgroups of o-Minimal Groups

Authors

Ya’acov Peterzil, Charles Steinhorn

Type: Article

Publication Date: 1999-06-01

Citations: 119

DOI: https://doi.org/10.1112/s0024610799007528

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Abstract

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.

Locations

Journal of the London Mathematical Society
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Definable compactness in o-minimal structures

2024-05-11

Pablo Andújar Guerrero

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of … We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types and definable downward directed families of closed sets.

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Definable compactness in definably complete locally o-minimal structures

2023-01-01

Masato Fujita

We demonstrate that And\'ujar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures. As an application, we show that … We demonstrate that And\'ujar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures. As an application, we show that a definably simple definable topological group which is regular, Hausdorff and definably compact as a definable topological space is either discrete or definably connected. We also study the definable quotient of definable continuous actions by definably compact definable topological groups.

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Definably Compact Sets in Definable Spaces Relative to O-minimal Structures

2020-07-12

Somayyeh Tari

A structure M is an o-minimal structure if every definable subset X of M is a finite union of intervals and points.Every o-minimal structure is a topological space. A definable … A structure M is an o-minimal structure if every definable subset X of M is a finite union of intervals and points.Every o-minimal structure is a topological space. A definable set X in M is called a definably compact set if every definable curve in it is completable in X. The definable compactness of a definable set X is equivalent to be closed and bounded.A idea of definable space generalizes theorem of Robson for semialgebraic spaces, A definablespace is not a definable set but it looks like the definable sets. Indeed definable spaces are constructed by the gluing of sets which are not definable but they are relativ to definable sets. In this paper we will recall the concept of definable space and introduce the notion of definable compactness in it. We show that a definable set X in the housdorff definable space is definable compact if and only if it is closed and bounded.

Definable compactness in definably complete locally o-minimal structures

2024-01-01

Masato Fujita

We demonstrate that Andújar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures.As an application, we show that a … We demonstrate that Andújar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures.As an application, we show that a definably simple definable topological group which is regular, Hausdorff and definably compact as a definable topological space is either discrete or definably connected.We also study the definable quotient of definable continuous actions by definably compact definable topological groups.

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MAXIMAL COMPACT SUBGROUPS IN THE O-MINIMAL SETTING

2013-04-11

Annalisa Conversano

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.

Definable rank, o-minimal groups, and Wiegold's problem

2023-01-01

Annalisa Conversano

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable … We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to o-minimal groups, of algebraic groups (over an algebraically closed field of characteristic 0) containing a Zariski-dense finitely generated subgroup. When the group is definably connected, its dimension provides an upper bound for the definable rank. The upper bound is often strict. For instance, this is always the case for 0-groups and, more generally, for solvable groups with torsion. We further prove that every perfect definable group is normally monogenic, generalizing the finite group case. This yields a positive answer to Wiegold's problem in the o-minimal setting.

Definable compactness in o-minimal structures

2025-03-05

Pablo Andújar Guerrero

Definable one-dimensional topologies in O-minimal structures

2019-05-31

Ya’acov Peterzil , Ayala Rosel

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On non-compact $p$-adic definable groups

2021-03-23

Will Johnson , Ningyuan Yao

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove … Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the Peterzil-Steinhorn theorem, in the special case of abelian groups. Let $G$ be an abelian group definable in a $p$-adically closed field $M$. If $G$ is not definably compact then there is a definable subgroup $H$ of dimension one which is not definably compact. In a future paper we will generalize this to non-abelian $G$.

TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01

Anand Pillay

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

Definable one dimensional topologies in o-minimal structures

2018-07-21

Ya’acov Peterzil , Ayala Rosel

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable … We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.

Definable one dimensional topologies in o-minimal structures

2018-01-01

Ya’acov Peterzil , Ayala Rosel

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Types, transversals and definable compactness in o-minimal structures.

2021-11-06

Pablo Andújar Guerrero

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded … Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in o-minimal expansions of ordered groups. Along the way we prove a parameter version for o-minimal theories of the connection between dividing and definable types known in the more general dp-minimal context [SS14], through an elementary proof that avoids the use of existing forking and VC literature. In particular we show that, if an $A$-definable family of sets has the $(p,q)$-property, for some $p\geq q$ with $q$ large enough, then the family admits a partition into finitely many subfamilies, each of which extends to an $A$-definable type.

One-dimensional subgroups and connected components in non-abelian $p$-adic definable groups

2023-01-01

William Johnson , Ningyuan Yao

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that … We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional definable subgroup which is not definably compact. This is a $p$-adic analogue of the Peterzil-Steinhorn theorem for o-minimal theories. Second, we show that if $G$ is a group definable over the standard model $\mathbb{Q}_p$, then $G^0 = G^{00}$. As an application, definably amenable groups over $\mathbb{Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when $G$ is a definable subgroup of a linear algebraic group, over any model.

Types, transversals and definable compactness in o-minimal structures

2021-01-01

Pablo Andújar Guerrero

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Discrete subgroups of locally definable groups

2013-03-22

Alessandro Berarducci , Mário J. Edmundo , Marcello Mamino

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Definable modules over definable rings without zero divisors in o-minimal structures

2020-01-01

Jaruwat Rodbanjong

In an o-minimal structure, every definable group admits a definable group manifold. Moreover, one can also show an analogue of this statement for definable rings. In this work, we prove … In an o-minimal structure, every definable group admits a definable group manifold. Moreover, one can also show an analogue of this statement for definable rings. In this work, we prove that every definable module over definable ring without zero divisors admits a definable module manifold. In addition, we also give the classification of definable modules over definable rings without zero divisors in o-minimal structures

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A REDUCTION TO THE COMPACT CASE FOR GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

2014-03-01

Annalisa Conversano

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.

A note on generic subsets of definable groups

2011-01-01

Mário J. Edmundo , Giuseppina Terzo

We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably … We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.

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One-dimensional definable topological spaces in o-minimal structures

2023-01-01

Pablo Andújar Guerrero , Margaret E. M. Thomas

We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) … We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical euclidean topology on definable sets, definable order topologies, definable quotient spaces and definable metric spaces. We use o-minimality to undertake their study in topological terms, focussing here in particular on spaces of dimension one. We present several results, given in terms of piecewise decompositions and existence of definable embeddings and homeomorphisms, for various classes of spaces that are described in terms of classical separation axioms and definable analogues of properties such as separability, compactness and metrizability. For example, we prove that all Hausdorff one-dimensional definable topologies are piecewise the euclidean, discrete, or upper or lower limit topology; we give a characterization of all one-dimensional, regular, Hausdorff definable topologies in terms of spaces that have a lexicographic ordering or a topology generalizing the Alexandrov double of the euclidean topology; and we show that, if the underlying structure expands an ordered field, then any one-dimensional Hausdorff definable topology that is piecewise euclidean is definably homeomorphic to a euclidean space. As applications of these results, we prove definable versions of several open conjectures from set-theoretic topology, due to Gruenhage and Fremlin, on the existence of a 3-element basis for regular, Hausdorff topologies and on the nature of perfectly normal, compact, Hausdorff spaces; we obtain universality results for some classes of Hausdorff and regular topologies; and we characterize when certain metrizable definable topologies admit a definable metric.

Topologizing Interpretable Groups in p-Adically Closed Fields

2023-11-01

Will Johnson

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar … We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group.

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Fundamental group in o-minimal structures with definable Skolem functions

2021-04-08

Bruno Dinis , Mário J. Edmundo , Marcello Mamino

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On Pillay's conjecture in the general case

2017-03-17

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

The six Grothendieck operations on o-minimal sheaves

2019-04-01

Mário J. Edmundo , Luca Prelli

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A note on $μ$-stabilizers in ACVF

2019-01-01

Jinhe Ye

We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and … We study $\mu$-stabilizers for groups definable in ACVF in the valued field sort. We prove that $\mathrm{Stab}^\mu(p)$ is an infinite unbounded definable subgroup of $G$ when $p$ is standard and unbounded. In the particular case when $G$ is linear algebraic, we show that $\mathrm{Stab}^\mu(p)$ is a solvable algebraic subgroup of $G$, with $\mathrm{dim}(\mathrm{Stab}^\mu(p))=\mathrm{dim}(p)$ when $p$ is $\mu$-reduced and unbounded.

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INTERPRETABLE SETS IN DENSE O-MINIMAL STRUCTURES

2018-12-01

Will Johnson

Abstract We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable … Abstract We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.

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Groups definable in two orthogonal sorts

2015-09-01

Alessandro Berarducci , Marcello Mamino

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Sheaves, Cosheaves and Applications

2013-01-01

Justin Curry

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial … This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that defines a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended pseudo-metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.

Lie-like decompositions of groups definable in o-minimal structures

2009-01-01

Annalisa Conversano

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study … There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.

Definable one-dimensional topologies in O-minimal structures

2019-05-31

Ya’acov Peterzil , Ayala Rosel

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Definability in the group of infinitesimals of a compact Lie group

2020-03-09

Martin Bays , Ya’acov Peterzil

We show that for G a simple compact Lie group, the infinitesimal subgroup G 00 is bi-interpretable with a real closed convexly valued field. We deduce that for G an … We show that for G a simple compact Lie group, the infinitesimal subgroup G 00 is bi-interpretable with a real closed convexly valued field. We deduce that for G an infinite definably compact group definable in an o-minimal expansion of a field, G 00 is bi-interpretable with the disjoint union of a (possibly trivial) ℚ-vector space and finitely many (possibly zero) real closed valued fields. We also describe the isomorphisms between such infinitesimal subgroups, and along the way prove that every definable field in a real closed convexly valued field R is definably isomorphic to R.

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Expansions of dense linear orders with the intermediate value property

2001-12-01

Chris Miller

Let ℜ be an expansion of a dense linear order ( R , <) without endpoints having the intermediate value property , that is, for all a, b ∈ R … Let ℜ be an expansion of a dense linear order ( R , <) without endpoints having the intermediate value property , that is, for all a, b ∈ R , every continuous (parametrically) definable function f : [ a, b ] → R takes on all values in R between f ( a ) and f(b) . Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of ( R , <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10). Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that ( R , <, *) is an ordered group. Then ( R , *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of ( R , *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of ( R , *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of ( R , *) definable in ℜ, then either G is dense and codense in R , or G contains an element u such that ( R , <, *, e, u, G ) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of ( R , *) (Theorem 2.3). Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

Pillay's conjecture and its solution—a survey

2010-06-07

Ya’acov Peterzil

Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the … Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups.

Linear Groups Definable in o-Minimal Structures

2002-01-01

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

On torsion-free groups in o-minimal structures

2005-10-01

Ya’acov Peterzil , Sergei Starchenko

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 .

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Topological groups, $\mu$-types and their stabilizers

2017-09-18

Ya’acov Peterzil , Sergei Starchenko

We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M … We consider an arbitrary topological group G definable in a structure \mathcal M , such that some basis for the topology of G consists of sets definable in \mathcal M . To each such group G we associate a compact G -space of partial types, S^{\mu}_G(M)=\{p_{\mu}\colon p\in S_G(M)\} which is the quotient of the usual type space S_G(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stab ^{\mu}(p) , which is the stabilizer of p_{\mu} . This group is nontrivial when p is unbounded in the sense of \mathcal M ; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of S^{\mu}_G(M) and its connection to the Samuel compactification of topological groups.

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Definable groups as homomorphic images of semi-linear and field-definable groups

2012-03-23

Pantelis E. Eleftheriou , Ya’acov Peterzil

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A descending chain condition for groups definable in <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>-minimal structures

2005-03-22

Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

Generic sets in definably compact groups

2007-01-01

Ya’acov Peterzil , Anand Pillay

A subset $X$ of a group $G$ is called left generic 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 left generic 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

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On central extensions and definably compact groups in o-minimal structures

2010-11-17

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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Definable one dimensional topologies in o-minimal structures

2018-07-21

Ya’acov Peterzil , Ayala Rosel

Locally definable subgroups of semialgebraic groups

2018-12-27

Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

O-minimal flows on nilmanifolds

2021-11-22

Ya’acov Peterzil , Sergei Starchenko

Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of UT(n,R), and let Γ be a lattice in G, with π:G→G∕Γ the quotient … Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of UT(n,R), and let Γ be a lattice in G, with π:G→G∕Γ the quotient map. For a semialgebraic X⊆G, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of π(X) in the compact nilmanifold G∕Γ. Our theorem describes cl(π(X)) in terms of finitely many families of cosets of real algebraic subgroups of G. The underlying families are extracted from X, independently of Γ. We also prove an equidistribution result in the case of curves.

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Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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.

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On central extensions and definably compact groups in o-minimal structures

2008-01-01

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple … We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).

Definable one dimensional structures in o-minimal theories

2010-10-31

Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

Groups, measures, and the NIP

2006-01-01

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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 a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

Computing o-minimal topological invariants using differential topology

2006-10-24

Ya’acov Peterzil , Sergei Starchenko

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of … We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

Strongly minimal groups in o-minimal structures

2018-10-03

Pantelis E. Eleftheriou , Assaf Hasson , Ya’acov Peterzil

We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a … We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.

Locally definable subgroups of semialgebraic groups

2019-10-07

Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil

We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: … We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].

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Cohomology of algebraic varieties over non-archimedean fields

2022-01-01

Pablo Cubides Kovacsics , Mário J. Edmundo , Jinhe Ye

Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf … Abstract We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K . For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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Valued fields, metastable groups

2019-07-23

Ehud Hrushovski , Silvain Rideau

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Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2021-08-19

Annalisa Conversano

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Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

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Characterizations of modules definable in o-minimal structures

2023-01-01

Jaruwat Rodbanjong , Athipat Thamrongthanyalak

<abstract>Let $ \mathfrak M $ be an o-minimal expansion of a densely linearly ordered set and $ (S, +, \cdot, 0_S, 1_S) $ be a ring definable in $ \mathfrak … <abstract>Let $ \mathfrak M $ be an o-minimal expansion of a densely linearly ordered set and $ (S, +, \cdot, 0_S, 1_S) $ be a ring definable in $ \mathfrak M $. In this article, we develop two techniques for the study of characterizations of $ S $-modules definable in $ \mathfrak M $. The first technique is an algebraic technique. More precisely, we show that every $ S $-module definable in $ \mathfrak M $ is finitely generated. For the other technique, we prove that every $ S $-module definable in $ \mathfrak M $ admits a unique definable $ S $-module manifold topology. As consequences, we obtain the following: (1) if $ S $ is finite, then a module $ A $ is isomorphic to an $ S $-module definable in $ \mathfrak M $ if and only if $ A $ is finite; (2) if $ S $ is an infinite ring without zero divisors, then a module $ A $ is isomorphic to an $ S $-module definable in $ \mathfrak M $ if and only if $ A $ is a finite dimensional free module over $ S $; and (3) if $ \mathfrak M $ is an expansion of an ordered divisible abelian group and $ S $ is an infinite ring without zero divisors, then every $ S $-module definable in $ \mathfrak M $ is definably connected with respect to the unique definable $ S $-module manifold topology.</abstract>

A semi-linear group which is not affine

2008-08-16

Pantelis E. Eleftheriou

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A note on generic subsets of definable groups

2011-01-01

Mário J. Edmundo , Giuseppina Terzo

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On minimal flows, definably amenable groups, and o-minimality

2015-12-30

Anand Pillay , Ningyuan Yao

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Splitting definably compact groups in o-minimal structures

2011-07-06

Marcello Mamino

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.

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On minimal flows and definable amenability in some distal NIP theories

2023-04-19

Ningyuan Yao , Zhentao Zhang

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One-dimensional groups over an o-minimal structure

1991-09-01

Vladimir Razenj

On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996-01-01

Margarita Otero , Ya’acov Peterzil , Anand Pillay

Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in … Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Groups of dimension two and three over o-minimal structures

1991-09-01

Ali Nesin , Anthony L. Pillay , V. Razenj

Zilber's conjecture for some o-minimal structures over the reals

1993-06-01

Ya’acov Peterzil

On o-minimal expansions of Archimedean ordered groups

1995-09-01

M. Laskowski , Charles Steinhorn

Abstract We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) … Abstract We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers . We then show that a definable function in an o-minimal expansion of enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of . Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group.

Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

Euler characteristic in semialgebraic and other o-minimal groups

1994-10-01

Adam Strzeboński