Boundedly Simple Groups Have Trivial Bounded Cohomology

Amenability and Acyclicity in Bounded Cohomology Theory

Johnson's characterization of amenable groups states that a discrete group $Γ$ is amenable if and only if $H_b^{n \geq 1}(Γ; V) = 0$ for all dual normed $\mathbb{R}[Γ]$-modules V. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective group homomorphism $ϕ\colon Γ\to K$ has amenable kernel H if and only if the induced inflation map $H^\bullet_b(K; V^H) \to H^\bullet_b(Γ; V)$ is an isometric isomorphism for every dual normed $\mathbb{R}[Γ]$-module V. In addition, we obtain an analogous characterization for the (smaller) class o…

有限群的>2度上同调中任意系数下的任意类可能在有限扩张中被消去

Arbitrary classes in >2-degree cohomology of a finite group with arbitrary coefficients may be trivialized in a finite extension

本文证明了有限群的高阶上同调类在有限扩张中可被消去。

Low degree bounded cohomology and l^2-invariants for negatively curved groups

We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a 'negatively curved' group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a Gromov hyperbolic group has to be virtually cyclic. Moreover, we show that any homomorphic image of an analogue of a higher rank lattices in a Gromov hyperbolic gr…

平凡线性循环集的上同调

Cohomology of Trivial Linear Cycle Sets

在H和I均为有限循环p-群（p为奇数，或p=2且H元素个数不超过4）的假设下，给出了平凡线性循环集H被交换群I扩张的完全分类，得到了所有可能扩张的显式参数化，并计算了所得线性循环集的基座和中心。

Infinite sums of Brooks quasimorphisms and cup products in bounded cohomology

We study a class of quasimorphisms of the free group that can be expressed as infinite sums of Brooks quasimorphisms with some nice properties. We then review Heuer's framework of decompositions developed in arXiv:1710.03193, and put these quasimorphisms in that context. This allows us to prove triviality of a number of cup products in the bounded cohomology of the free group with trivial real coefficients. Finally, we introduce free products of decompositions and relate them to Rolli's quasimorphisms of free products. A technical result concerning these, which remains an open question, implie…

Cup Product in Bounded Cohomology of the Free Group

The theory of bounded cohomology of groups has many applications. A key open problem is to compute the full bounded cohomology $H_b^n(F, R)$ of a non-abelian free group $F$ with trivial real coefficients. It is known that $H_b^n(F,R)$ is trivial for $n=1$ and uncountable dimensional for $n=2,3$, but remains unknown for any $n \geq 4$. For $n=4$, one may construct classes by taking the cup product $α\cup β\in H_b^4(F, R)$ between two $2$-classes $α, β\in H^2_b(F, R)$. However, we show that all such cup products are trivial if $α$ and $β$ are classes induced by the quasimorphisms defined by Broo…

On the vanishing ranges for the cohomology of finite groups of Lie type II

The computation of the cohomology for finite groups of Lie type in the describing characteristic is a challenging and difficult problem. In earlier work, the authors constructed an induction functor which takes modules over the finite group of Lie type to modules for the ambient algebraic group G. In particular this functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in the earlier work to determine the first non-trivial cohomology class when the underlying root system is of type A_{n} or C_{n}. In this paper the authors extend these results t…

Trivial Massey Product in Bounded Cohomology

We show that in the bounded cohomology of non-abelian free groups the Massey triple product is always trivial when the second factor is represented by the coboundary of a decomposable quasi-morphism. We also show that in the bounded cohomology of a negatively curved compact Riemannian manifold the Massey triple product is always trivial when the second factor is represented by an exact differential form.

On the low dimensional cohomology groups of the IA-automorphism group of a free group of rank three

In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of $\mathrm{IA}_3$, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the low…

Polynomial Cohomology and Polynomial Maps on Nilpotent Groups

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the expo…

欧几里得空间与圆盘的变换群的有界上同调

The bounded cohomology of transformation groups of Euclidean spaces and discs

本文证明了 $\mathbb{R}^n$ 的保向同胚群和微分同胚群在所有正则性下均有界无环，这是对非紧支撑变换群有界上同调的首次完整计算，并据此得出平坦丛的许多示性类是无界的。

Cohomology, Extensions and Automorphisms of Skew Braces

The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial braces is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.

A note on bounded-cohomological dimension of discrete groups

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.

On the Third-Degree Continuous Cohomology of Simple Lie Groups

We show that the class of connected, simple Lie groups that have non-vanishing third-degree continuous cohomology with trivial $\mathbb{R}$-coefficients consists precisely of all simple complex Lie groups and of $\widetilde{\mathrm{SL}_2(\mathbb{R})}$.

第二有界上同调为零的有限生成单左可序群

Finitely generated simple left orderable groups with vanishing second bounded cohomology

本文证明一类有限生成单左可序群在平凡实系数与整系数下第二有界上同调均为零，给出首个非不可指示的此类群例，并解答Navas的ICM问题。

横截测度群胚的有界上同调归纳

Bounded cohomological induction for transverse measured groupoids

本文为横截测度群胚建立可测有界上同调的归纳同构，推广Eckmann-Shapiro同构，并证明其与群连续有界上同调的等价性，进而给出非平凡有界上同调群的首次显式计算。

Dyer-Lashof operations on Tate cohomology of finite groups

Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in…

Vanishing up to the rank in bounded cohomology

We establish the vanishing for non-trivial unitary representations of the bounded cohomology of SL_d up to the rank. It holds more generally for uniformly bounded representations on superreflexive spaces. The same results are obtained for lattices. We also prove that the real bounded cohomology of any lattice is invariant in the same range.

First cohomology of pure mapping class groups of big genus one and zero surfaces

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.

The Johnson homomorphism and the second cohomology of IA_n

Let F_n be the free group on n generators. Define IA_n to be group of automorphisms of F_n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA_n to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of IA_n. A descending central series of IA_n is given by the subgroups K_n^(i) which act trivially on F_n/F_n^(i+1), the free rank n, degree i nilpotent group. It is a conjecture of Andreadakis that K_n^(i) is equal to the lower central series of IA_n; indeed K_n^(2) is known to…

对称群与交错群乘积上同调的essential商

Essential quotients of the cohomology of products of symmetric and alternating groups

本文计算对称群与交错群乘积的模二上同调关于可忽略元素的商，回答Serre提出的问题，并基于其近期结果，用于域的上同调不变量研究。

A formula for the second cohomology of two-step nilpotent groups

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.

The integral cohomology of the group of loops

Let PSigma_n denote the group that can be thought of either as the group of motions of the trivial n-component link or the group of symmetric automorphisms of a free group of rank n. The integral cohomology ring of PSigma_n is determined, establishing a conjecture of Brownstein and Lee.

Central extensions and bounded cohomology

It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a…

Transgression in bounded cohomology and a conjecture of Monod

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.