Nuclear dimension and virtually polycyclic groups

Caleb Eckhardt, Jianchao Wu
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Abstract

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on $C_0(X)$-algebras.
核维度和实际多环群
我们证明了实际上多环群(扭曲的)群 C* 代数的核维度是有限的。这促使我们提出了一个将群 C* 代数的有限核维度与有限赫希长度联系起来的猜想,然后我们对一类超越了实际上多环情况的基本可调和群进行了验证。我们还提出了一个关于有限分解秩的平行猜想,并得到了类似的结果。我们的方法在很大程度上依赖于希尔施伯格和第二位作者最近在$C_0(X)$数组上的近似无穷群作用的研究。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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