Maximal Compact Subgroups in the O-Minimal Setting

Conversano, Annalisa

doi:10.1142/s0219061313500049

Cited by 8 publications

(16 citation statements)

References 21 publications

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“…[, Proposition 5.5]). The papers show that this condition holds for some large classes of definable groups, such as semisimple or linear. Moreover, Proposition Let G be a definable definably connected group.…”

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 67%

“…The work done in shows that every definable group G contains a maximal definable torsion‐free subgroup, however the existence of a maximal definable, definably compact one is not guaranteed: indeed it is equivalent to G having a definable compact‐torsion‐free decomposition (cf. [, Proposition 5.5]). The papers show that this condition holds for some large classes of definable groups, such as semisimple or linear.…”

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 98%

“…Our first step is to find a suitable decomposition of the group G by means of its subgroups whose definable flows can be computed. The following results are due to Conversano and hold for any model of an o‐minimal expansion of

sans-serifRCF

, the theory of real closed ordered fields: Definition Let G be a definable group. We say that G has a definable compact‐torsion‐free decomposition if for some definable, definably compact

K < G

and a definable torsion‐free

H < G

we have

G = K \cdot H

and

K \cap H = {e}

.…”

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 99%

“…If

G = K H

is such a decomposition, then K and H are the maximal definable subgroups which are respectively compact and torsion‐free. The work done in shows that every definable group G contains a maximal definable torsion‐free subgroup, however the existence of a maximal definable, definably compact one is not guaranteed: indeed it is equivalent to G having a definable compact‐torsion‐free decomposition (cf. [, Proposition 5.5]).…”

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 99%

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Definable topological dynamics and real Lie groups

Jagiella

2015

Mathematical Logic Qtrly

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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 G=KH, we give a description of minimal subflows and the Ellis group of its universal definable flow SG(double-struckR) in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to NG(H)∩K(double-struckR). This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to G/G00. We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort double-struckR and a sort for an abelian group.

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Section: Compact‐torsion‐free Decompositionmentioning

confidence: 67%

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 98%

sans-serifRCF

, the theory of real closed ordered fields: Definition Let G be a definable group. We say that G has a definable compact‐torsion‐free decomposition if for some definable, definably compact

K < G

and a definable torsion‐free

H < G

we have

G = K \cdot H

and

K \cap H = {e}

.…”

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 99%

“…If

G = K H

Section: Compact‐torsion‐free Decompositionmentioning

confidence: 99%

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Definable topological dynamics and real Lie groups

Jagiella

2015

Mathematical Logic Qtrly

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“…We now turn to the structural theory developed by Conversano in [3] and [4] to describe an arbitrary group G. Definition 5.3. We say that G has a definable compact-torsion-free decomposition if there are definable subgroups K, H ă G with K X H " t1 G u such that K is definably compact, and H is torsion-free.…”

Section: Another Results Comes From [5]mentioning

confidence: 99%

The Ellis Group Conjecture and Variants of Definable Amenability

Jagiella¹

2018

J. symb. log.

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We consider definable topological dynamics for N IP 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.2010 Mathematics Subject Classification. 03C99, 54H20 (primary), and 03C64 (secondary).

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

Conversano

2021

Journal of Algebra

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Maximal Compact Subgroups in the O-Minimal Setting

Cited by 8 publications

References 21 publications

Definable topological dynamics and real Lie groups

Definable topological dynamics and real Lie groups

The Ellis Group Conjecture and Variants of Definable Amenability

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

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