Ursula Hamenstaedt
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
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Cocycles and positive functionals in higher cohomology
研究高阶单位上同调中正功能算子与余弦的对应关系，推广Gelfand-Naimark-Segal构造并探讨正功能算子的消失条件。
This paper studies Moore's measurable cohomology theory for locally compact groups and Polish modules. An elementary dimension-shifting argument is used to show that all classes in that theory have representatives with considerable extra topological structure beyond measurability. Using this, for certain target modules one can also construct a direct comparison map with a different cohomology theory for topological groups defined by Segal, and show that this map is an isomorphism.
Let $E$ be a virtually central extension of the group $G$ by a finitely generated abelian group $A$. We show that $E$ carries a biautomatic structure if and only if $G$ has a biautomatic structure $L$ for which the cohomology class of the extension is represented by an $L$-regular cocycle. Moreover, a cohomology class is $L$-regular if some multiple of it is or if its restriction to some finite index subgroup is. We also show that the entire second cohomology of a Fuchsian group is regular, so any virtually central extension is biautomatic. In particular, if the fundamental group of a Seifert …
In [5], the notion of polynomial cocycles is used to give an expression for the second cohomology of T-groups with coefficients in a torsion-free nilpotent module. We make this expression concrete in the case of a T-group G of nilpotency class <=2 and coefficients in a trivial G-module, using a Lie algebra associated to the group. This approach allows us to construct explicit cocycles representing the elements of the second cohomology group.
Given a topological group $ G $ and a Hausdorff topological group $ A $ on which $ G $ acts continuously and compatibly with the group operation of $ A $, we study the set of continuous cocycles of $ G $ with value in $ A $. This set is a function space and can be endowed with several topologies. By imposing a suitable function space topology on the set of cocycles of $ G $ with value in $ A $, we propose a topological study of this set, and we prove, as our first main result, that if $ A $ is a compact group having a presentation as an inverse limit of compact and Hausdorff topological groups…
We compute the group cohomology of $32Γ_3f$, a certain group of order 32. For this we construct explicit cocycle representatives of the cohomology generators. We thus lay to rest a discrepancy between several published computations of this cohomology ring.
Recently E. R. Alvares, M. M. Alves and M. J. Redondo introduced a cohomology for a group $G$ with values in a module over the partial group algebra $K_{\mathrm{par}}(G)$, which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action $α$ of $G$ on a (unital) algebra $\mathcal{A}$ we consider $\mathcal{A}$ as a $K_{\mathrm{par}}(G)$-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in $\mathcal{A}$. The problem is reduced to an …
Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$
本文在低次数情形显式构造了双曲空间等距群乘积及 $\mathrm{SL}(3,\mathbb{K})$ 的 Furstenberg 边界上的非交错与交错上闭链，并由此证明相应可测有界上同调比较映射在次数 3 处的单射性。
To characterize categorical constraints - associativity, commutativity and monoidality - in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups gives non-abelian cohomology. The categorification - functor from groups to monoidal categories - provides the correspondence between the respective parity (quasi)complexes and allows to interpret 1-cochains as functors, 2-cocycles - monoidal structures, 3-cocycles - associators. The cohomology spaces H3, H2, H1, H0 correspond as usual to quasi-extensions, exten…
Symmetric 2-cocycles with values in $\mathbb{C}^\times$
本文研究了有限群中具有值在$\mathbb{C}^ imes$的对称2-余循环，证明了存在一个阶为64的群具有非平凡余同调类的对称2-余循环。
Computing Galois cohomology of a real linear algebraic group
本文提出并实现了计算实数域上一般线性代数群第一个伽罗瓦上同调集合的算法，可输出 1-上循环列表，并能对给定上循环判定等价性并给出实现等价的元素。
We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.
We verify Shalom's conjecture for the simple real-rank-one Lie group Sp(n ,1) for any n: i.e. we show that it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. We provide two different proofs. Both approaches crucially use results on uniformly bounded representations by Michael Cowling. The first approach is quite abstract: it uses an automatic-properness result of Shalom and requires almost no computations. The second approach is explicit: we deduce the properness of cocycles from the non-continuity of a critical case of the S…
In this article we study cocycles of discrete countable groups with values in l^2(G) and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in l^2(G) and answer several questions from [CTV]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first l^2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.
We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.
Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology
本文证明紧群作用下的李群值相对连续上循环空间构成解析子流形，并分解为纤维丛，由此得出上同调空间离散，并应用于态射空间与射影表示空间的流形结构。
We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,π)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.
Let $G$ be a finite simple group and $k$ be an algebraically closed field of prime characteristic dividing the order of $G$. We show that for all $2$-cocycles $α\in Z^2(G;k^\times)$, the first Hochschild cohomology group of the twisted group algebra $HH^1(k_αG)$ is nonzero.
We prove that for a weakly mixing algebraic action $σ: G\curvearrowright(X,ν)$, the $n$-cohomology group $H^n(G\curvearrowright X; \mathbb{T})$, after quotienting out the natural subgroup $H^n(G,\mathbb{T})$, contains $H^n(G,\widehat{X})$ as a natural subgroup for $n=1$. If we further assume the diagonal actions $σ^2, σ^4$ are $\mathbb{T}$-cocycle superrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, then the above also holds true for $n=2$. Applying it for principal algebraic actions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ is torsion free as an abelian group when $G$…
We prove that the cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres. In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely.
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