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## Filters

##
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Semisimple stable and superstable groups

1989
*
Annals of Pure and Applied Logic
*

*A*

*subgroup*H

*of*

*a*

*group*G is

*type*-dejiruzbk or (/\-

*definable*) if universe

*of*H is the intersection

*of*

*a*family

*of*at most IL1

*definable*subsets G) and the CC unction

*of*the &

*define*

*a*

*subgroup*in every ... In fact, most

*of*our

*type*-

*definable*arise in this way. 0 hand, Lascar and Berline in [6] at, again when G is

*stable*, the intersection

*of*any family

*of*deF$able al to the intersection

*of*at most IL1

*definable*... Cl Let H be

*a*

*type*-

*definable*

*subgroup*

*of*

*a*1 TI+-saturated

*stable*

*group*G. ...

##
###
Stable groups and expansions of (Z,+,0)
[article]

2017
*
arXiv
*
pre-print

In particular, this answers

arXiv:1601.05692v3
fatcat:3654u6px2zeadpagjjmwamvzkm
*a*question*of*Marker*on**stable*expansions*of*the*group**of*integers by sets*definable*in Presburger arithmetic. ... We show that if G is*a*sufficiently saturated*stable**group**of*finite weight with no infinite, infinite-index, chains*of**definable**subgroups*, then G is superstable*of*finite U-rank. ... We are grateful to Rizos Sklinos and Erik Walsberg for their comments*on*an earlier draft. ...##
###
ON MAXIMAL STABLE QUOTIENTS OF DEFINABLE GROUPS IN NIP THEORIES

2018
*
Journal of Symbolic Logic (JSL)
*

AbstractFor G

doi:10.1017/jsl.2017.26
fatcat:mnngwjv4dngexbprprdnknrhma
*a**group**definable*in*a*saturated model*of**a*NIP theory T, we prove that there is*a*smallest*type*-*definable**subgroup*H*of*G such that the quotient G / H is*stable*. ... This generalizes the existence*of*G00, the smallest*type*-*definable**subgroup**of*G*of*bounded index. ... Suppose T has N IP , and let G be*a**definable**group*(over ∅). Let H be*a**type*-*definable**subgroup**of*G such that G/H is*stable*. ...##
###
A note on superstable groups

2005
*
Journal of Symbolic Logic (JSL)
*

It is proved that all

doi:10.2178/jsl/1120224734
fatcat:fqikxdbalzhkxerdtsvhyygm5m
*groups**of*finite U-rank that have the descending chain condition*on**definable**subgroups*are totally transcendental. ...*A*corollary is that any*stable**group*that is*definable*in an o -minimal structure is totally transcendental*of*finite Morley rank. ... Poizat's theorem [8] that every*type*-*definable**subgroup**of**a**stable**group*is the intersection*of**definable**subgroups*, together with the d.c.c., implies that H is*a**definable**subgroup**of*G/H n . ...##
###
On maximal stable quotients of definable groups in NIP theories
[article]

2016
*
arXiv
*
pre-print

For G

arXiv:1606.01182v1
fatcat:k7y7l5rtibajtcnefijchroqzy
*a**group**definable*in an NIP theory we prove that there is*a*smallest*type*-*definable**subgroup*H*of*G such that the quotient G/H is*stable*. ... This generalizes the existence*of*G^00, the smallest*type*-*definable**subgroup**of*G*of*bounded index. ... Suppose T has NIP , and let G be*a**definable**group*(over ∅). Let H be*a**type*-*definable**subgroup**of*G such that G/H is*stable*. ...##
###
On Variants of CM-triviality
[article]

2012
*
arXiv
*
pre-print

We show that, under this condition,

arXiv:1205.6916v2
fatcat:km5zqtzhwjabrfpzwqmvq3zg3m
*a**stable*field is internal to the family, and*a**group**of*finite Lascar rank has*a*normal nilpotent*subgroup*such that the quotient is almost internal to the family. ... We introduce*a*generalization*of*CM-triviality relative to*a*fixed invariant collection*of*partial*types*, in analogy to the Canonical Base Property*defined*by Pillay, Ziegler and Chatzidakis which generalizes ... Hrushovski and Pillay have shown [9] that in*a**stable**one*-based*group*,*definable*sets are finite unions*of*cosets*of*acl eq (∅)-*definable**subgroups*. Furthermore, the*group*is abelian-by-finite. ...##
###
On variants of CM-triviality

2012
*
Fundamenta Mathematicae
*

We show that, under this condition,

doi:10.4064/fm219-3-4
fatcat:pybsajulyzhldewjxyzs5cnyqm
*a**stable*field is internal to the family, and*a**group**of*finite Lascar rank has*a*normal nilpotent*subgroup*such that the quotient is almost internal to the family. ... We introduce*a*generalization*of*CM-triviality relative to*a*fixed invariant collection*of*partial*types*, in analogy to the Canonical Base Property*defined*by Pillay, Ziegler and Chatzidakis which generalizes ... Hrushovski and Pillay have shown [9] that in*a**stable**one*-based*group*,*definable*sets are finite unions*of*cosets*of*acl eq (∅)-*definable**subgroups*. Furthermore, the*group*is abelian-by-finite. ...##
###
Book Review: Stable groups

2002
*
Bulletin of the American Mathematical Society
*

This result is followed by

doi:10.1090/s0273-0979-02-00953-9
fatcat:bka54xadg5eavlvx47pbipt2ea
*a*series*of*propositions*on*descending chain conditions*on**subgroups**of**stable**groups*. ... We*define*G 0 (φ) to be the intersection*of*all*subgroups**of*G*of*the form H*a*:= {g ∈ G : φ(g,*a*)}*of*finite index in G. For*a**stable**group*G, the*group*G 0 (φ) is necessarily*definable*. ... Thomas Scanlon University*of*California, Berkeley E-mail address: scanlon@math.berkeley.edu ...##
###
DEFINABLE GROUPS IN DCFA

2019
*
Revista de Matemática: Teoría y Aplicaciones
*

Then, we study

doi:10.15517/rmta.v26i2.33639
fatcat:uzkkiqb4nvfm5e2noryooyy5p4
*one*-basedeness, stability and*stable*embeddability*of*abelian*definable**groups*. ... In this paper we study*definable*abelian*groups*in*a*model*of*DCFA. First we prove that such*a**group*is embeddable*on*an algebraic*group*. ... By 1-basedness and quantifier-free ω-stability, we know that if X ⊂*A*♯ is quantifier-free*definable*, then X is*a*Boolean combination*of*cosets*of*quantifier-free*definable**subgroups**of**A*♯ . ...##
###
Complete reducibility: Variations on a theme of Serre
[article]

2021
*
arXiv
*
pre-print

By results

arXiv:2004.14604v3
fatcat:tkpmsvy535g2tisbufbfsraji4
*of*Serre and Bate–Martin–Röhrle, the usual notion*of*G-complete reducibility can be re-framed as*a*property*of*an action*of**a**group**on*the spherical building*of*the identity component*of*G. ... In this note, we unify and extend various concepts in the area*of*G-complete reducibility, where G is*a*reductive algebraic*group*. ... Acknowledgements: We would like to thank Michael Bate for helpful discussions*on*the material*of*this note. ...##
###
Complete reducibility: variations on a theme of Serre

2021
*
Manuscripta mathematica
*

By results

doi:10.1007/s00229-021-01318-2
fatcat:cpuffrdarnfo3h6qxpvnzci3zy
*of*Serre and Bate–Martin–Röhrle, the usual notion*of*G-complete reducibility can be re-framed as*a*property*of*an action*of**a**group**on*the spherical building*of*the identity component*of*G. ... AbstractIn this note, we unify and extend various concepts in the area*of*G-complete reducibility, where G is*a*reductive algebraic*group*. ... To view*a*copy*of*this licence, visit http://creativecommons.org/ licenses/by/4.0/. ...##
###
Book Review: Simple groups of finite Morley rank

2010
*
Bulletin of the American Mathematical Society
*

In the mid-1990s

doi:10.1090/s0273-0979-10-01287-5
fatcat:7wkxy6qbw5fc5pp2vrjmhd7xue
*a*view was consolidated that this project falls into four cases*of*different flavour: even*type*, mixed*type*, odd*type*, and degenerate*type*. ... Morley showed that in an uncountably categorical theory, all*definable*sets have ordinal-valued Morley rank. In particular, such theories are*stable*(in fact, ω-*stable*). ... points*of*an algebraic*group**defined*over*a**stable*field K. ...##
###
Page 4133 of Mathematical Reviews Vol. , Issue 92h
[page]

1992
*
Mathematical Reviews
*

*A*series

*of*applications

*of*

*type*-

*definable*

*subgroups*

*of*

*a*

*stable*

*group*is given. ... (PL-PAW)

*On*

*type*

*definable*

*subgroups*

*of*

*a*

*stable*

*group*. Notre Dame J. Formal Logic 32 (1991), no. 2, 173-187. ...

##
###
Definable Groups in DCFA
[article]

2019
*
arXiv
*
pre-print

First we prove that such

arXiv:1904.11586v1
fatcat:zwuzq2mwvfftnm4cwvohv4ueem
*a**group*is embeds*on*an algebraic*group*. Then we study 1-basedeness, stability and*stable*embeddability*of*abelian*definable**groups*. ... Hrushovski proved that the theory*of*difference-differential fields*of*characteristic zero has*a*model-companion. We denote it DCFA. In this paper we study*definable**groups*in*a*model*of*DCFA. ... with*a*′ over M , b is interalgebraic with b ′ over M and c is interalgebraic with c ′ over M Then there is*a**type*-*definable*over M*subgroup*G 1*of*bounded index in G, and*a**type*-*definable*over M*subgroup*...##
###
Model Theory and Groups
[article]

2021
*
arXiv
*
pre-print

This is

arXiv:2109.03911v1
fatcat:lkzf7qgd65e75kb2ng2zmvanmq
*a*largely expository paper about how*groups*arise or are*of*interest in model theory. ... Included are the following topics: classifying*groups**definable*in specific structures or theories and the relation to algebraic*groups*,*groups**definable*in*stable*, simple and NIP theories,*definable*compactifications ...*Stable**groups**Stable**group*theory is an adaptation*of*the above machinery to the context*of**definable*homogeneous spaces (*a*transitive*definable*action*of**a**definable**group*G*on**a**definable*set X), although ...
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