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Steve Jackson  One of the best experts on this subject based on the ideXlab platform.

countable Abelian Group actions and hyperfinite equivalence relations
Inventiones Mathematicae, 2015CoAuthors: Steve JacksonAbstract:An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations \(E_n\) where all \(E_n\)equivalence classs are finite. In this article we establish the following theorem: if a countable Abelian Group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructions and analysis of Borel marker sets and regions in the space \(2^{{\mathbb {Z}}^{<\omega }}.\) This technique is also applied to a problem of finding Borel chromatic numbers for invariant Borel subspaces of \(2^{{\mathbb {Z}}^n}\).

Countable Abelian Group actions and hyperfinite equivalence relations, preprint
2015CoAuthors: Su Gao, Steve JacksonAbstract:Abstract. An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations En where all Enequivalence classs are finite. In this article we establish the following theorem: if a countable Abelian Group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructions and analysis of Borel marker sets and regions in the space 2Z <ω. This technique is also applied to a problem of finding Borel chromatic numbers for invariant Borel subspaces of 2
Nathaniel Stapleton  One of the best experts on this subject based on the ideXlab platform.

the balmer spectrum of the equivariant homotopy category of a finite Abelian Group
Inventiones Mathematicae, 2019CoAuthors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a finite Abelian Group A, we determine the Balmer spectrum of \({\mathrm {Sp}}_A^{\omega }\), the compact objects in genuine Aspectra. This generalizes the case \(A={\mathbb {Z}}/p{\mathbb {Z}}\) due to Balmer and Sanders (Invent Math 208(1):283–326, 2017), by establishing (a corrected version of) their \(\hbox {log}_p\)conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixedpoints and establish a generalization of Kuhn’s blueshift theorem for Tateconstructions (Kuhn in Invent Math 157(2):345–370, 2004).

the balmer spectrum of the equivariant homotopy category of a finite Abelian Group
arXiv: Algebraic Topology, 2017CoAuthors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a finite Abelian Group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{BalmerSanders}, by establishing (a corrected version of) their log$_p$conjecture for Abelian Groups. We work out the consequences for the chromatic type of fixedpoints. We also establish a generalization of Kuhn's blueshift theorem for Tateconstructions \cite{kuhn}.

the balmer spectrum of the equivariant homotopy category of a finite Abelian Group
arXiv: Algebraic Topology, 2017CoAuthors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel StapletonAbstract:For a finite Abelian Group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{BalmerSanders}, by establishing (a corrected version of) their log$_p$conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixedpoints and establish a generalization of Kuhn's blueshift theorem for Tateconstructions \cite{kuhn}.
Daniel V Tausk  One of the best experts on this subject based on the ideXlab platform.

a locally compact non divisible Abelian Group whose character Group is torsion free and divisible
Canadian Mathematical Bulletin, 2013CoAuthors: Daniel V TauskAbstract:It was claimed by Halmos in 1944 that if G is a Hausdorff locally compact topological Abelian Group and if the character Group of G is torsion free, then G is divisible. We prove that such a claim is false by presenting a family of counterexamples. While other counterexamples are known, we also present a family of stronger counterexamples, showing that even if one assumes that the character Group of G is both torsion free and divisible, it does not follow that G is divisible. Departamento de Matematica, Universidade de Sao Paulo, Brazil email: tausk@ime.usp.br Received by the editors March 3, 2010; revised January 9, 2011. Published electronically July 9, 2011. AMS subject classification: 22B05. 1

a locally compact non divisible Abelian Group whose character Group is torsion free and divisible
arXiv: General Topology, 2010CoAuthors: Daniel V TauskAbstract:It has been claimed by Halmos in [Comment on the real line, Bull. Amer. Math. Soc., 50 (1944), 877878] that if G is a Hausdorff locally compact topological Abelian Group and if the character Group of G is torsion free then G is divisible. We prove that such claim is false, by presenting a family of counterexamples. While other counterexamples are known (see [D. L. Armacost, The structure of locally compact Abelian Groups, 1981]), we also present a family of stronger counterexamples, showing that even if one assumes that the character Group of G is both torsion free and divisible, it does not follow that G is divisible.
Archil Gulisashvili  One of the best experts on this subject based on the ideXlab platform.

rearrangements of functions on a locally compact Abelian Group and integrability of the fourier transform
Journal of Functional Analysis, 1997CoAuthors: Archil GulisashviliAbstract:Abstract We find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact Abelian Group. These classes consist of all functions for which the Fourier transform belongs to a given Lorentz space on the dual Group. Different special cases of the problems considered in this paper have been originally studied by Hardy, Littlewood, Hewitt, Ross, Cereteli, and the author.
M Woronowicz  One of the best experts on this subject based on the ideXlab platform.

A torsionfree Abelian Group of finite rank exists whose quotient Group modulo the square subGroup is not a nilGroup
Quaestiones Mathematicae, 2017CoAuthors: R R Andruszkiewicz, M WoronowiczAbstract:The first example of a finite rank torsionfree Abelian Group A such that the quotient Group of A modulo the square subGroup of A is not a nilGroup is indicated (in both cases of associative and g...

a torsion free Abelian Group exists whose quotient Group modulo the square subGroup is not a nil Group
Bulletin of The Australian Mathematical Society, 2016CoAuthors: R R Andruszkiewicz, M WoronowiczAbstract:The first example of a torsionfree Abelian Group such that the quotient Group of modulo the square subGroup is not a nilGroup is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian Groups, nil modulo a subGroup, need not have nil quotient Group’, Publ. Math. Debrecen 27 (1980), 127–130] is given for torsionfree Groups. A new method of constructing indecomposable nilGroups of any rank from to is presented. Ring multiplications on pure subGroups of the additive Group of the ring of adic integers are investigated using only elementary methods.

some new results for the square subGroup of an Abelian Group
Communications in Algebra, 2016CoAuthors: R R Andruszkiewicz, M WoronowiczAbstract:The complete description of the square subGroup of a torsion Abelian Group and an elementary construction of a mixed Abelian Group (A, +, 0), such that the quotient Group of A modulo the square subGroup □A is not a nilGroup, are given (also for the associative case). Some A. M. Aghdam's results concerning square subGroups for the associative case are proven. The relationship between square subGroups for both the cases of associative and general rings is partially investigated.