完全断开局部紧密群作用于建筑物的韦尔不变式

Ilaria Castellano, Bianca Marchionna, Thomas Weigel
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引用次数: 0

摘要

在某些情况下,作用于局部有限建筑物的紧凑生成的完全互不相连局部紧凑群的不变式可以通过代表建筑物类型的考斯特群的不变式来方便地描述。对于作用于建筑物的某些完全互不相连的局部紧凑群,我们建立并收集了几个结果,例如,合理离散同调维数(参见 Thm.A)、平秩(参见 Thm.C)和端数(参见 Cor.K)。此外,对于一个任意紧凑生成的完全断开局部紧凑群,我们用它的同调群来表达末端数(参见 Thm.J )。此外,我们推广了 F. Haglund 和 F. Paulin 的一个结果,证明考斯特群 $(W,S)$ 的可视群图分解可以用来从类型 $(W,S)$ 的建筑物中构造树。我们利用后一个结果来证明,所有对建筑物起室反作用的$\sigma$-compact totally disconnected locally compact群,都可以相应地分解为类型为$(W,S)$的任何可见群分解图(参见定理F和定理G)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Weyl-invariants of totally disconnected locally compact groups acting cocompactly on buildings
In several instances, the invariants of compactly generated totally disconnected locally compact groups acting on locally finite buildings can be conveniently described via invariants of the Coxeter group representing the type of the building. For certain totally disconnected locally compact groups acting on buildings, we establish and collect several results concerning, for example, the rational discrete cohomological dimension (cf. Thm. A), the flat-rank (cf. Thm. C) and the number of ends (cf. Cor. K). Moreover, for an arbitrary compactly generated totally disconnected locally compact group, we express the number of ends in terms of its cohomology groups (cf. Thm. J). Furthermore, generalising a result of F. Haglund and F. Paulin, we prove that visual graph of groups decompositions of a Coxeter group $(W,S)$ can be used to construct trees from buildings of type $(W,S)$. We exploit the latter result to show that all $\sigma$-compact totally disconnected locally compact groups acting chamber-transitively on buildings can be decomposed accordingly to any visual graph of groups decomposition of the type $(W,S)$ (cf. Thm. F and Cor. G).
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