Stable finiteness properties of infinite discrete groups
Stable finiteness properties of infinite discrete groups
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无限离散群的稳定有限性
DOI：
10.1112/topo.12035
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发表时间：
2016
影响因子：
1.1
通讯作者：
Irakli Patchkoria
中科院分区：
文献类型：
--
作者：
No'e B'arcenas;D. Degrijse;Irakli Patchkoria
设G是无限离散群。G的真作用的分类空间是真G-CW复形X，使得不动点集XH对G的所有有限子群H是可压缩的。本文考虑了真G-谱范畴中真作用的分类空间的稳定类比，并研究了它的有限性质。我们研究了G何时允许有限或有限类型的真作用的稳定分类空间，并将这些条件与真G-谱的同伦范畴中球谱的紧性以及G的有限子群的Weyl群的经典有限性质联系起来。最后，如果群G是几乎无挠的，我们还证明了群G的正常行为的稳定分类空间的最小可能维度与G的虚拟上同调维度重合，从而首次给出了群的虚拟上同调维度的几何解释。
Let G be an infinite discrete group. A classifying space for proper actions of G is a proper G ‐CW complex X such that the fixed point sets XH are contractible for all finite subgroups H of G . In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper G ‐spectra and study its finiteness properties. We investigate when G admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper G ‐spectra and to classical finiteness properties of the Weyl groups of finite subgroups of G . Finally, if the group G is virtually torsion‐free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of G , thus providing the first geometric interpretation of the virtual cohomological dimension of a group.