We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group $S(M)$ and the group of positive scalar curvature metrics $P(M)$ for an oriented manifold $M$.

We define a class of groups called “polynomially full groups” for which the upper bound and the lower bound we derive are the same.We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator $K$-theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups.

中文翻译：

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算子$ K $-理论的有限部分的秩的界

我们得出了有限生成群的算子$ K $-理论组的最大部分和归约$ C ^ * $-代数的有限部分的秩的下限和上限。下限基于组中有限阶元素的多项式增长的共轭类的数量。上限基于组中扭转元素的数量。我们使用下界为结构组$ S（M）$和有向标量$ M $的正标量曲率指标$ P（M）$给出下界。

我们定义了一个称为“多项式完全组”的组，其上界和下界是相同的。我们表明，该多项式完全组包含了几乎所有的幂等组。例如，对于有限生成的阿贝尔群，对称群和二面体群，我们给出了算子$ K $-理论群的有限部分秩的明确公式。