A survey on groups definable in o-minimal structures

Otero, Margarita

doi:10.1017/cbo9780511735219.006

Cited by 23 publications

(14 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now review some earlier results, mainly about definably simple and definable semisimple groups (for a survey of definable groups in o-minimal structures, see Otero's [20]). …”

Section: Preliminary Resultsmentioning

confidence: 99%

See 1 more Smart Citation

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

Hrushovski¹,

Peterzil²,

Pillay³

2011

Journal of Algebra

View full text Add to dashboard Cite

We prove several structural results on definable, 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, (ii) (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 deducing the full compact domination conjecture for definably compact groups from the semisimple and commutative cases which were already settled. 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.

show abstract

“…We now review some earlier results, mainly about definably simple and definable semisimple groups (for a survey of definable groups in o-minimal structures, see Otero's [20]). …”

Section: Preliminary Resultsmentioning

confidence: 99%

“…(i) We use induction on dim G. If dim G = 1 then G is abelian (see [20,Proposition 3.1]) and therefore divisible. We consider the general case.…”

Section: Finite Extensions Of O-minimal Groupsmentioning

confidence: 99%

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

Hrushovski¹,

Peterzil²,

Pillay³

2011

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…When Fact 6 applies, one can strengthen Lemma 5(b) as follows. Again the following statement is valid in groups definable in an o-minimal structure, because they cannot contain an infinite increasing chain of definably connected subgroups (by the existence of a well behaved notion of dimension [24,Corollary 2.4]).…”

Section: Cartan Subgroups and Carter Subgroupsmentioning

confidence: 99%

Cartan subgroups of groups definable in o-minimal structures

Baro

Jaligot

Otero³

2013

J. Inst. Math. Jussieu

Self Cite

View full text Add to dashboard Cite

show abstract

“…We refer to [28] for a general introduction to o-minimal structures, and to [19] for groups definable in o-minimal structures. We will frequently point out analogy to the finite Morley rank case in [3].…”

Section: Introductionmentioning

confidence: 99%

Commutators in groups definable in o-minimal structures

Baro

Jaligot

Otero

2012

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

show abstract

A survey on groups definable in o-minimal structures

Cited by 23 publications

References 51 publications

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

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

Cartan subgroups of groups definable in o-minimal structures

Commutators in groups definable in o-minimal structures

Contact Info

Product

Resources

About