Geometry of growth: approximation theorems for $L^2$ invariants
Geometry of growth: approximation theorems for
$L^2$ invariants
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增长几何：$L^2$ 不变量的近似定理
DOI：
10.1007/s002080050190
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发表时间：
1997
影响因子：
1.4
通讯作者：
M. Farber
中科院分区：
文献类型：
--
作者：
M. Farber
本文研究了L^2 $-拓扑不变量的有限维近似问题。我们得到了L“uck定理的推广，处理了塔的单层正规覆盖。我们证明逼近定理，建立同调不变量之间的关系，对应于无限维表示和序列的有限维表示，假设他们的归一化字符收敛。此外，我们发现了一个逼近定理剩余有限$p$-群（$p$是一个素数），其中我们使用的同调系数在有限域$\fp$。
我们认为序列的有限维平坦丛的增长尺寸的增长过程的例子。我们研究了一个具有Dixandian型迹的von Neumann范畴，它可以描述增长过程的渐近不变量。我们引入了一个新的不变量的扭转对象，扭转维数。我们表明，扭转尺寸出现在一般作为一个额外的校正项的逼近定理，它消失在一些算术性假设。我们还表明，扭转尺寸允许建立非平凡的Grothendieck组的扭转对象。
In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of L\"uck, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite $p$-groups ($p$ is a prime), where we use the homology with coefficients in a finite field $\fp$.
We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish non-triviality of the Grothendieck group of torsion objects.