Finite decomposition rank for virtually nilpotent groups

Caleb Eckhardt; Elizabeth Gillaspy; Paul McKenney

doi:10.1090/tran/7453

Finite decomposition rank for virtually nilpotent groups

Caleb Eckhardt
, Elizabeth Gillaspy
, Paul McKenney

Mathematical Sciences

Miami University

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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·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.

Original language	English
Pages (from-to)	3971-3994
Number of pages	24
Journal	Transactions of the American Mathematical Society
Volume	371
Issue number	6
DOIs	https://doi.org/10.1090/tran/7453
State	Published - Mar 15 2019

Funding

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). Received by the editors July 21, 2017, and, in revised form, October 30, 2017. 2010 Mathematics Subject Classification. Primary 46L05; Secondary 20F19, 46L35, 46L55, 46L80. 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).

Funders	Funder number
Simons Foundation
SFB 878

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10.1090/tran/7453

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N2 - 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·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.

AB - 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·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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Finite decomposition rank for virtually nilpotent groups

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