论文标题
在非Archimedean领域的规范舒适性和有限的同胞学
Normed amenability and bounded cohomology over non-Archimedean fields
论文作者
论文摘要
我们研究了完全断开的本地紧凑型组的连续有限的共同体，并在非架构的$ k $中具有系数。为了捕获诱导真实有限的共同体消失的经典不适当的特征，我们介绍了规范的$ k $ - 不明性的概念，我们证明了代数的表征。这意味着规范为$ k $ nomable的群体是本地椭圆形的，它与其离散有限的$ p $ subquotients的订单相关联，其中$ k $ nomebable group的规范是$ k $的残基领域的特征。此外，我们证明了离散组的有界 - 人体学表征。 代数表征表明，规范$ k $ - 不典型性是一个非常限制的条件，因此有限的共同体学表明，应该有很多具有丰富有限共同体的小组，具有琐碎的$ K $系数。我们通过研究比较图的注射率和溢流性来探讨这种直觉，为此，可以进行令人惊讶的一般性陈述。在其中，我们表明，如果$ k $具有正性特征或其残基字段具有特征0，那么比较图在所有程度上都是配件的。如果$ k $是$ \ mathbb {q} _p $的有限扩展名，我们将组的准晶体分类并将其与其亚组结构相关联。对于离散组，我们表明合适的有限条件暗示比较图是同构。 与实际情况形成鲜明对比的比较图通常是同构的动机，是通过移动到拓扑空间给出的。我们表明，在非架构领域，有界的共同体学是艾伦贝格（Eilenberg）（斯蒂恩罗德（Eilenberg）意义上的一种同谋理论，除了添加性公理的较弱版本外，这对于有限的脱节工会来说是等效的。特别是有一个梅耶 - 越野鸟序列。
We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field $K$. To capture the features of classical amenability that induce the vanishing of real bounded cohomology, we introduce the notion of normed $K$-amenability, of which we prove an algebraic characterization. It implies that normed $K$-amenable groups are locally elliptic, and it relates an invariant, the norm of a $K$-amenable group, to the order of its discrete finite $p$-subquotients, where $p$ is the characteristic of the residue field of $K$. Moreover, we prove a bounded-cohomological characterization for discrete groups. The algebraic characterization shows that normed $K$-amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial $K$ coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either $K$ has positive characteristic or its residue field has characteristic 0, then the comparison map is injective in all degrees. If $K$ is a finite extension of $\mathbb{Q}_p$, we classify quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism. A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg--Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer--Vietoris sequence.
