Model theoretic connected components of finitely generated nilpotent groups

Bowler, Nathan; Chen, Cong; Gismatullin, Jakub

doi:10.2178/jsl.7801170

Cited by 4 publications

(10 citation statements)

References 11 publications

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“…Thus, the absolute connectedness of H follows from Proposition 3. 4 We finish with a discussion on Steinberg symbols. Note that a given field may have many different orders.…”

Section: More Precisely Supposementioning

confidence: 98%

On model-theoretic connected components in some group extensions

Gismatullin

Krupiński

2015

J. Math. Log.

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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 SL 2 (R) (intepreted 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. We also obtain a variant of the example of Conversano and Pillay for SL 2 (Z) instead of SL 2 (R) , which (as most of our examples) was not accessible by the previously known methods.

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“…Thus, the absolute connectedness of H follows from Proposition 3. 4 We finish with a discussion on Steinberg symbols. Note that a given field may have many different orders.…”

Section: More Precisely Supposementioning

confidence: 98%

On model-theoretic connected components in some group extensions

Gismatullin

Krupiński

2015

J. Math. Log.

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“…If G is a definable group in M , and a small set, recall the following well-known subgroups of , so-called model-theoretic connected components : , the intersection of all A -definable subgroups of with finite index, , the smallest A -type-definable subgroup of with bounded index, and , the smallest A -invariant subgroup of with bounded index. We refer to [2, 5] for the properties of the connected components which we are going to use and explain below. Clearly .…”

Section: Preliminariesmentioning

confidence: 99%

“…Suppose for a contradiction that G is not of finite exponent. Then, by [2, Lemma 4.9], there is a homomorphism from G to with dense image. Such a homomorphism is an element of of infinite order, a contradiction.⊣…”

Section: Preliminariesmentioning

confidence: 99%

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Bohr Compactifications of Groups and Rings

Gismatullin¹,

Jagiella²,

Krupiński³

2022

J. symb. log.

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We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$ , the continuous Heisenberg group ${\mathrm {UT}}_3({\mathbb {R}})$ , and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

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“…A , the smallest A-invariant subgroup of Ḡ with bounded index. We refer to [Gis11,BCG13] for the properties of the connected components which we are going to use and explain below. Clearly Ḡ000…”

Section: Preliminariesmentioning

confidence: 99%

Bohr compactifications of groups and rings

Gismatullin¹,

Jagiella²,

Krupiński³

2020

Preprint

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We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group UT3pZq, the continuous Heisenberg group UT3pRq, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

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Model theoretic connected components of finitely generated nilpotent groups

Cited by 4 publications

References 11 publications

On model-theoretic connected components in some group extensions

On model-theoretic connected components in some group extensions

Bohr Compactifications of Groups and Rings

Bohr compactifications of groups and rings

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