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Nonstandard hulls of locally uniform groups

100%
Goldbring I.
Fundamenta Mathematicae
|
2013
|
tom 220
|
nr 2
93-118
EN
We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxemburg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used to construct the hull. We use the nonstandard hull construction to give a nonstandard characterization of Enflo's notion of groups that are uniformly free from small subgroups. We prove that our nonstandard hull is locally isomorphic to Pestov's nonstandard hull for Banach-Lie groups. We also give some examples of infinite-dimensional Lie groups that are locally uniform.
2
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The fundamental group of a locally finite graph with ends-a hyperfinite approach

63%
Goldbring I., Sisto A.
Fundamenta Mathematicae
|
2016
|
tom 232
|
nr 1
21-39
EN
The end compactification |Γ| of a locally finite graph Γis the union of the graph and its ends, endowed with a suitable topology. We show that π₁(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups. We also show how to recover the standard description of π₁(|Γ|) given by Diestel and Sprüssel (2011). Finally, we give some applications of our result, including a short proof that certain loops in |Γ| are non-nullhomologous.
3
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Existentially closed II₁ factors

51%
Farah I., Goldbring I., Hart B., Sherman D.
Fundamenta Mathematicae
|
2016
|
tom 233
|
nr 2
173-196
EN
We examine the properties of existentially closed ($𝓡 ^{ω}$-embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ($𝓡 ^{ω}$-embeddable) II₁ factor is approximately inner to prove that Th(𝓡) is not model-complete. We also show that Th(𝓡) is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th(𝓡).
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