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AbstractWe find an explicit form of the inverse isomorphism from Shapiro’s lemma in terms of inhomogeneous cocycles and apply it to construct special nonsplit coverings of groups with a unique conjugacy class of involutions. [ABSTRACT FROM AUTHOR]

We prove that a strengthened version of Minač–Tân's Massey Vanishing Conjecture holds true for fields with a finite number of square classes whose maximal pro-2 Galois group is of elementary type (as defined by I. Efrat). In particular, this proves Minač–Tân's Massey Vanishing Conjecture for Pythagorean fields with a finite number of square classes and their finite extensions. [ABSTRACT FROM AUTHOR]

Given a group G and a partial factor set σ of G, we introduce the twisted partial group algebra κ par σ G , which governs the partial projective σ -representations of G into algebras over a field κ. Using the relation between partial projective representations and twisted partial actions we endow κ par σ G with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of κ par σ G. Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product A ∗ Θ G , involving the Hochschild homology of A and the partial homology of G, where Θ is a unital twisted partial action of G on a κ -algebra A with a κ -based twist. An analogous third quadrant cohomological spectral sequence is also obtained. [ABSTRACT FROM AUTHOR]

We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincaré dualities for them and prove a stabilization property for the infinite series of reflection groups. [ABSTRACT FROM AUTHOR]

We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points. [ABSTRACT FROM AUTHOR]

For a non-compact finite thickness building whose Davis apartment is an orientable pseudomanifold, we compute the supremum of the set of p > 1 such that its top dimensional reduced ℓ p -cohomology is nonzero. We adapt the non-vanishing assertion of this result to any finite thickness building using the Bestvina realization. Using similar techniques, we generalize bounds obtained by Clais on the conformal dimension of some Gromov-hyperbolic buildings to any such building. [ABSTRACT FROM AUTHOR]

Given a finite group G, we introduce "encoding pairs," which are a pair of G-modules M and M ′ equipped with a shifted natural isomorphism between the cohomological functors H • (G , Hom Z (M , −)) and H • (G , Hom Z (M ′ , −)) . Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere. [ABSTRACT FROM AUTHOR]

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg–Tran–Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane. [ABSTRACT FROM AUTHOR]

In (Xu, J Pure Appl Algebra 212:2555–2569, 2008), a LHS-spectral sequence for target regular extensions of small categories is constructed. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Słomińska's spectral sequence for the cohomology of regular EI-categories. We show that many well-known spectral sequences related to the homology decompositions of finite groups, centric linking systems, and the orbit category of fusion systems can be obtained as the LHS-spectral sequence of an extension. [ABSTRACT FROM AUTHOR]

Let p be a prime. We prove that certain amalgamated free pro-p products of Demushkin groups with pro-p-cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro-p group, and thus do not occur as maximal pro-p Galois groups of fields containing a root of 1 of order p. We show that other cohomological obstructions which are used to detect pro-p groups that are not maximal pro-p Galois groups—the quadraticity of Z / p Z -cohomology and the vanishing of Massey products—fail with the above pro-p groups. Finally, we prove that the Minač–Tân pro-p group cannot give rise to a 1-cyclotomic oriented pro-p group, and we conjecture that every 1-cyclotomic oriented pro-p group satisfy the strong n-Massey vanishing property for n = 3 , 4 . [ABSTRACT FROM AUTHOR]

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring. [ABSTRACT FROM AUTHOR]

Let G be a group. Among many other results, it is well known that the Schur multiplicator of G (the second integral homology group of G) is torsion-free for free Abelian groups of finite rank. We consider here some groups, the multiplicator which is torsion-free. As a simple observation, we find that the multiplicator of a divisible nilpotent group is torsion-free. We also prove that the Schur multiplicator of an upper unitriangular group G of order 4 and also of G / γ 3 (G) is torsion-free. Some other examples of groups, the multiplicator which is torsion-free are given. [ABSTRACT FROM AUTHOR]

We prove a statement concerning hyperlinearity for central extensions of property (T) groups in the presence of flexible HS-stability, and more generally, weak ucp-stability. Notably, this result is applied to show that if Sp 2 g (Z) is flexibly HS-stable, then there exists a non-hyperlinear group. Further, the same phenomenon is shown to hold generically for random groups sampled in Gromov’s density model, as well as all infinitely presented property (T) groups. This gives new directions for the possible existence of a non-hyperlinear group. Our results yield Hilbert–Schmidt analogues for Bowen and Burton’s work relating flexible P-stability of PSL n (Z) and the existence of non-sofic groups [10]. [ABSTRACT FROM AUTHOR]

The aim of the paper is to give a full classification of factorizations of groups in terms of descent cohomology (pointed) sets introduced in Mesablishvili (Trans. A. Razmadze Math. Inst. 173(2), 137–155, 2019). We show that descent cohomology includes Serre's non-abelian group cohomology as a special case. This enables us to generalize Serre's theory further to include monoids. [ABSTRACT FROM AUTHOR]

Eilenberg and Ganea proved the equality cd (Γ) = cat Γ for discrete groups Γ . We explore the similar equality cat (ϕ) = cd (ϕ) for group homomorphisms. In particular, we prove it for homomorphisms ϕ : Γ → Λ of any torsion free finitely generated nilpotent group Γ to an arbitrary group Λ . We construct a counterexample ψ : G → H , cat (ψ) > cd (ψ) , with geometrically finite groups. [ABSTRACT FROM AUTHOR]

We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we give a sharp bound for the size of the Schur multiplier of special p-group of all ranks. We also improve existing bounds for the size of Schur multiplier of p-groups (p ≠ 2 , 3) of class c ≥ 3 , and for p-groups of coclass r. [ABSTRACT FROM AUTHOR]

For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology. [ABSTRACT FROM AUTHOR]

Rota–Baxter operators for groups were recently introduced by Guo, Lang, and Sheng. Bardakov and Gubarev showed that with each Rota–Baxter operator, one can associate a skew brace. Skew braces on a group G can be characterised in terms of certain gamma functions from G to its automorphism group Aut (G) that are defined by a functional equation. For the skew braces obtained from a Rota–Baxter operator, the corresponding gamma functions take values in the inner automorphism group Inn (G) of G. In this paper, we give a characterisation of the gamma functions on a group G, with values in Inn (G) , that come from a Rota–Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota–Baxter operator. For gamma functions that can be obtained from a Rota–Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits. [ABSTRACT FROM AUTHOR]

We consider the Euler characteristics $\chi (M)$ of closed, orientable, topological $2n$ -manifolds with $(n-1)$ -connected universal cover and a given fundamental group G of type $F_n$ . We define $q_{2n}(G)$ , a generalised version of the Hausmann-Weinberger invariant [19] for 4–manifolds, as the minimal value of $(-1)^n\chi (M)$ . For all $n\geq 2$ , we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of G. As an application, we obtain new restrictions for nonabelian finite groups arising as fundamental groups of rational homology 4–spheres.

Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

Suppose that G is a finite group and k is a field of characteristic p > 0 . We consider the complete cohomology ring E M ∗ = ∑ n ∈ Z Ext ^ kG n (M , M) . We show that the ring has two distinguished ideals I ∗ ⊆ J ∗ ⊆ E M ∗ such that I ∗ is bounded above in degrees, E M ∗ / J ∗ is bounded below in degree and J ∗ / I ∗ is eventually periodic with terms of bounded dimension. We prove that if M is neither projective nor periodic, then the subring of all elements in negative degrees in E M ∗ is a nilpotent algebra. [ABSTRACT FROM AUTHOR]

We prove that the cup product of Δ-decomposable quasimorphisms, Brooks quasimorphisms or Rolli quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial. [ABSTRACT FROM AUTHOR]

Let G be the Klein four-group and let k be an arbitrary field of characteristic 2. A classification of indecomposable k G -modules is known. We calculate the relative cohomology groups H χ i (G , N) for every indecomposable k G -module N, where χ is the set of proper subgroups in G. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in H χ * (G , k) are trivial. [ABSTRACT FROM AUTHOR]

For a prime number ℓ we say that an oriented pro- ℓ group (G , θ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal θ -abelian quotient π G , θ ab : G → G (θ) is a free pro- ℓ group contained in the Frattini subgroup of G. We show that oriented pro- ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat's Elementary Type Conjecture implies a positive answer to Positselski's version of Bogomolov's Conjecture on maximal pro- ℓ Galois groups of a field K in case that K × / (K ×) ℓ is finite. Secondly, it is shown that for an H ∙ -quadratic oriented pro- ℓ group (G , θ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map d 2 2 , 1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4). [ABSTRACT FROM AUTHOR]

We show via $\ell^2$ -homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat–Tits building. [ABSTRACT FROM AUTHOR]

The two-circulant core (TCC) construction for Hadamard matrices uses two sequences with almost perfect autocorrelation to construct a Hadamard matrix. A research problem of K. Horadam asks whether such matrices are cocyclic. Using techniques from the theory of permutation groups, we prove that the order of a cocyclic TCC matrix coincides with the order of a Hadamard matrix of Paley type, of Sylvester type or certain multiples of these orders. We show that there exist cocyclic TCC Hadamard matrices at all allowable orders ⩽ 1000 with at most one exception. Of the four families of TCC matrices known in the literature, we establish that two are cocyclic, prove that one is not cocyclic, and leave one undecided. The undecided family consists of matrices of 2-power order; we show that these are inequivalent to the Sylvester matrices. As a generalisation of the TCC construction, we introduce quadruple-circulant core (QCC) Hadamard matrices; our results give a complete description of the orders that admit cocyclic QCC Hadamard matrices. [ABSTRACT FROM AUTHOR]

Suppose F is a finite extension of Qp, G is the group of F‐points of a connected reductive F‐group, and I1 is a pro‐p‐Iwahori subgroup of G. We construct two spectral sequences relating derived functors on mod‐p representations of G to the analogous functors on Hecke modules coming from pro‐p‐Iwahori cohomology. More specifically: (1) using results of Ollivier–Vignéras, we provide a link between the right adjoint of parabolic induction on pro‐p‐Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a 'Poincaré duality spectral sequence' relating duality on pro‐p‐Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules Hi(I1,π). [ABSTRACT FROM AUTHOR]

In this short survey article, we aim to provide an up to date information on the progress made towards Schurs exponent property and related conjectures. We also mention the connection between Schurs exponent property and Noether's Rationality problem. [ABSTRACT FROM AUTHOR]

Let G be a group, let H be a subgroup of G and let Or(G) be the orbit category. In this paper we extend the definition of the relative group homology the- ories of the pair (G, H) defined by Adamson and Takasu to have coefficients in an Or(G)-module. There is a canonical comparison homomorphism defined by Cisneros- Molina and Arciniega-Nevrez from Takasu's theory to Adamson's theory. We give a necessary and sufficient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism. [ABSTRACT FROM AUTHOR]