Abelian Group

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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, 2015
    Co-Authors: Steve Jackson
    Abstract:

    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
    2015
    Co-Authors: Su Gao, Steve Jackson
    Abstract:

    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 En-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 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, 2019
    Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel Stapleton
    Abstract:

    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 (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 fixed-points and establish a generalization of Kuhn’s blue-shift theorem for Tate-constructions (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, 2017
    Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel Stapleton
    Abstract:

    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{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for Abelian Groups. We work out the consequences for the chromatic type of fixed-points. We also establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \cite{kuhn}.

  • the balmer spectrum of the equivariant homotopy category of a finite Abelian Group
    arXiv: Algebraic Topology, 2017
    Co-Authors: Tobias Barthel, Niko Naumann, Justin Noel, Thomas Nikolaus, Markus Hausmann, Nathaniel Stapleton
    Abstract:

    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{Balmer-Sanders}, by establishing (a corrected version of) their log$_p$-conjecture for Abelian Groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions \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, 2013
    Co-Authors: Daniel V Tausk
    Abstract:

    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 e-mail: 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, 2010
    Co-Authors: Daniel V Tausk
    Abstract:

    It has been claimed by Halmos in [Comment on the real line, Bull. Amer. Math. Soc., 50 (1944), 877-878] 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.

M Woronowicz - One of the best experts on this subject based on the ideXlab platform.