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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Finite decomposition rank for virtually nilpotent groups


Authors: Caleb Eckhardt, Elizabeth Gillaspy and Paul McKenney
Journal: Trans. Amer. Math. Soc. 371 (2019), 3971-3994
MSC (2010): Primary 46L05; Secondary 20F19, 46L35, 46L55, 46L80
DOI: https://doi.org/10.1090/tran/7453
Published electronically: December 3, 2018
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Abstract: We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group $ G$ is bounded by $ 2\cdot h(G)!-1$, where $ h(G)$ is the Hirsch length of $ G.$ This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.


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Additional Information

Caleb Eckhardt
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio, 45056
Email: eckharc@miamioh.edu

Elizabeth Gillaspy
Affiliation: Department of Mathematical Sciences, University of Montana, 32 Campus Drive # 0864, Missoula, Montana, 59812-0864
Email: elizabeth.gillaspy@mso.umt.edu

Paul McKenney
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio, 45056
Email: mckennp2@miamioh.edu

DOI: https://doi.org/10.1090/tran/7453
Received by editor(s): July 21, 2017
Received by editor(s) in revised form: October 30, 2017
Published electronically: December 3, 2018
Additional Notes: The first author was partially supported by a grant from the Simons Foundation.
The second author was primarily supported by the Deutsches Forschungsgemeinschaft via SFB 878 (awarded to the Universität Münster, Germany).
Article copyright: © Copyright 2018 American Mathematical Society