Finite decomposition rank for virtually nilpotent groups

Eckhardt, Caleb; Gillaspy, Elizabeth; McKenney, Paul

doi:10.1090/tran/7453

Finite decomposition rank for virtually nilpotent groups
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by Caleb Eckhardt, Elizabeth Gillaspy and Paul McKenney PDF

Trans. Amer. Math. Soc. 371 (2019), 3971-3994 Request permission

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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