We investigate small $p$-groups with cohomology rings of depth higher than predicted by Duflot's theorem. In these groups, a sampling would suggest several naive conjectures about the degrees of the additional regular sequence elements. We arrive at counterexamples to the idea that all cohomology rings exceeding Duflot's bound have degree 2 regular sequence elements past Duflot's bound as well as to the idea that regular sequences can be pulled back, preserving degrees, along a field extension.
We compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups.
In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof in…
We introduce a way of describing cohomology of the symmetric groups with coefficients in Specht modules over Z or F_p. We study i-th-degree cohomology for i in {0,1,2}. The focus lies on the isomorphism type of second-degree cohomology of integral Specht modules. Unfortunately, only in few cases can we determine this exactly. In many cases we obtain only some information about the prime divisors of the cohomology group's order. The most important tools we use are the Zassenhaus algorithm, the Branching Rules, Bockstein type homomorphisms, and the results from [Burichenko et al., 1996].
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 $\widetilde{\mathrm{SL}_2(\mathbb{R})}$.
For $M$ a compact Riemannian manifold Brandenbursky and Marcinkowski constructed a transfer map $H_b^*(π_1(M))\to H_b^*(Homeo_{vol,0}(M))$ and used it to show that for certain $M$ the space $\overline{EH}_b^3(Homeo_{vol,0}(M))$ is infinite-dimensional. Kimura adapted the argument to $Diff_{vol}(D^2,\partial D^2)$. We extend both results to the higher degrees $\overline{EH}_b^{2n}$, $n\geq 1$. We also show that for certain $M$ the ordinary cohomology $H^*(Homeo_{vol,0}(M))$ is non-trivial in all degrees. In our computations we view the transfer map as being induced by a coupling of groups.
The aim of the current paper is to explore the implications on the group $G$ of the non-vanishing of the cohomology in degree one of one of its representation $π$, given some mixing conditions on $π$. In one direction, harmonic cocycles are used to show that the FC-centre should be finite (for mildly mixing unitary representations). Next, for any subgroup $H<G$, $H$ will either be "small", almost-malnormal or $π_{|H}$ also has non-trivial cohomology in degree one (in this statement, "small", reduced vs unreduced cohomology and unitary vs generic depend on the mixing condition). The notion of q…
Let $G$ be a reductive group defined over an algebraically closed field of characteristic $0$ such that the Dynkin diagram of $G$ is the disjoint union of diagrams of types $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$. We show that the degree $3$ unramified cohomology of the classifying space of $G$ is trivial. In particular, combined with articles by Merkurjev \cite{Mer17} and the author \cite{Baek}, this completes the computations of degree $3$ unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type.
It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL_2 over S-integers.
We characterize those Lie groups, and algebraic groups over a local field of characteristic zero, whose first reduced L^p-cohomology is zero for all p>1, extending a result of Pansu. As an application, we obtain a description of Gromov-hyperbolic groups among those groups. In particular we prove that any non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local field of zero characteristic is quasi-isometric to a 3-regular tree. We also extend the study to semidirect products of a general locally compact group by a cyclic group acting by contracting automorphisms.
D. J. Benson conjectures that the Castelnuovo-Mumford regularity of a group cohomology ring is always zero. More generally he conjectures that the cohomology ring always has a system of parameters satisfying a property he calls very strong quasi-regular. Using computer calculations we find that the more general conjecture holds for all groups of order less than 256.
It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of…
If G is a countable discrete group acting linearly on a finite-dimensional vector space over any topological field, then the groups of coboundaries are closed for the product topology in all degrees, and hence the cohomology is reduced in all degrees. This is deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces.
The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results showing its triviality are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; when $p \in ]1,\infty[$ and $G$ is Liouville (in particular, of intermediate growth). This is done by answering a question of Pansu assuming the graph satisfies an isoperimetric profile. Namely, the triviality of the reduced $\ell^p$-cohomology is equivalent to the absence of non-co…
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.
Given a group satisfying sufficient finiteness properties, we discuss a group algebra criterion for vanishing of all its cohomology groups with unitary coefficients in a certain degree.
We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is non-zero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree one Hochschild cohomology as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn is a consequence of a general formula that makes more precise a theorem of our prev…
