论文标题
多项式生长群的次要较高不变和循环共同体
Secondary Higher Invariants and Cyclic Cohomology for Groups of Polynomial Growth
论文作者
论文摘要
我们证明，如果$γ$是一组多项式生长，那么该组代数上的每个离域循环共生都具有多项式生长的代表。因此，对于每个离域的Cocyle，我们都定义了Lott Delabalizate ETA不变性的更高类似物，并证明了其对可逆差异操作员的收敛性。我们还使用Xie和Yu的确定地图构造来证明，如果$γ$是多项式增长的，那么在离域循环cocyles和$ k $ - $ c^*$ - 代数 - 代数次级次要上级不变的较高不变的款之间有一个明确定义的配对。当这个$ k $的理论类是可逆差分运算符的较高Rho不变性的类别时，我们表明这种配对恰恰是Lott Delabalized ETA不变性的上述高级类似物。作为这种对等的应用，我们提供了一个$ m $的较高的Atiyah-Patodi-Singer索引定理，这是一个带有边界的紧凑型旋转歧管，配备了正标准度量$ g $，并具有基本的$γ=π_1（m）$，这是有限生成和多元化生长的。
We prove that if $Γ$ is a group of polynomial growth then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocyle we thus define a higher analogue of Lott's delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if $Γ$ is of polynomial growth then there is a well defined pairing between delocalized cyclic cocyles and $K$-theory classes of $C^*$-algebraic secondary higher invariants. When this $K$-theory class is that of a higher rho invariant of an invertible differential operator we show this pairing is precisely the aforementioned higher analogue of Lott's delocalized eta invariant. As an application of this equivalence we provide a delocalized higher Atiyah-Patodi-Singer index theorem given $M$ is a compact spin manifold with boundary, equipped with a positive scalar metric $g$ and having fundamental group $Γ=π_1(M)$ which is finitely generated and of polynomial growth.
