# Finite decomposition rank for virtually nilpotent groups

## TL;DR

This paper proves that the C*-algebras of finitely generated virtually nilpotent groups have bounded decomposition rank, extending previous results and enabling classification via Elliott invariants under certain conditions.

## Contribution

It establishes a bound on the decomposition rank of group C*-algebras for virtually nilpotent groups and extends quasidiagonality results to inductive limits of such groups.

## Key findings

- Decomposition rank of finitely generated virtually nilpotent group C*-algebras is bounded by 2·h(G)! - 1.
- Inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras.
- C*-algebras from these groups satisfying the UCT are classified by their Elliott invariants.

## 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.