B-有界上同调及其应用
B-Bounded cohomology and applications
摘要：
一个字长为（G，L）的离散群对于一个边界类B是B-等同调的，如果从B-有界上同调到普通上同调（复系数）的比较映射是同构的;如果对于任意系数也是如此，则它是强B-等同调的。本文建立了保证强B-等余同调的一些基本条件。特别地，当所有的加权G-敏感Dehn函数都是B-有界的时，我们证明了$FP^{\infty}$群G的强B-等余同调性.这样的群包括所有的B-异步可梳群;而且，这样的群的类在作用于非循环复形的群的构造下是封闭的。我们还提供了比较映射不能是内射的例子，以及满射，并给出了一个例子的可解群的二次第一德恩函数，但指数第二德恩函数。最后，给出了群关于子群的B-有界群的上同调的相关理论。相对isocohologicality确定在一个新的概念，相对Dehn功能和相对$FP^\infty$属性的群体就收集的子群。在相对双曲群和群的可展复形的背景下，给出了计算群的B-有界群的上同调的应用。
英文摘要：
A discrete group with word-length (G,L) is B-isocohomological for a bounding classes B if the comparison map from B-bounded cohomology to ordinary cohomology (with complex coefficients) is an isomorphism; it is strongly B-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong B-isocohomologicality. In particular, we show strong B-isocohomologicality for an $FP^{\infty}$ group G if all of the weighted G-sensitive Dehn functions are B-bounded. Such groups include all B-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of B-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relative $FP^\infty$ property for groups with respect to a collection of subgroups. Applications for computing B-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.
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