Related papers: $\Phi$-Harmonic Functions on Discrete Groups and F…

We prove that the continuous group cohomology groups of a locally profinite group $ G $ with coefficients in a smooth $ k $-representation $ \pi $ of $ G $ are isomorphic to the $ \mathrm{Ext}$-groups $ \mathrm{Ext}^i_G(\mathbb{1},\pi) $…

In this paper, we obtain a new proof result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $\ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and…

The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups…

Reduced cohomology motivates to look at harmonic functions which satisfy certain gradient conditions. If $G$ is a direct product of two infinite groups or a (FC-central)-by-cyclic group, then there are no harmonic functions with gradient in…

We prove that for generalised partitions of unity ${\phi_i \mid i \in I}$ and coverings $\mathfrak{U}:={\phi_i^{-1} (R \setminus {0}) \mid i \in I}$ of a topological space $X$ the cohomology of abstract $\mathfrak{U}$-local cochains…

By an $\ell$-group $G$ we mean a lattice-ordered abelian group. This paper is concerned with the category $\FP$ of finitely presented {\it unital} $\ell$-groups, those $\ell$-groups having a distinguished order-unit $u$. Using the duality…

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We use the infinitary logic to show…

Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is…

Let $\Phi$ be a real valued function of one real variable, let $L$ denote an elliptic second order formally self-adjoint differential operator with bounded measurable coefficients, and let $P$ stand for the Poisson operator for $L$. A…

Let $\phi: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $\phi$ is dense. We show that all such homomorphisms arise as completions of $G$ with respect to…

We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the F-cohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every…

In this article, we study the isomorphism problem for the algebras of $\Phi-$Pseudofunctions and $\Phi-$Pseudomeasures, denoted by $PF_\Phi(G)$ and $PM_\Phi(G),$ respectively. More precisely, for a certain class of Young functions $\Phi,$…

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq…

To each finitely generated group $G$, we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of $G$. If $G$ is finitely presented, our invariant is closely related to the Dehn function of $G$, but provides more information…

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$, endowed with a rational self-map $\Phi$. Let $\alpha\in G(K)$ and let $\Gamma\subseteq G(K)$ be a finitely generated subgroup. We show that the set…

Let $G$ be a discrete group. We give a decomposition theorem for the Hochschild cohomology of $\ell^1(G)$ with coefficients in certain $G$-modules. Using this we show that if $G$ is commutative-transitive, the canonical inclusion of bounded…

Let $G$ be a reductive algebraic group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $\sigma$ be an automorphism of $G$ of order $\ell$ -- a prime number -- with $\ell\neq p$. Let $\Pi$ be a finite length…

We prove a homotopy invariance result for the first cohomology group of the special unitary group $\mathrm{SU}_3(F[t])$ with coefficients in irreducible representations of $\mathrm{PGL}_2(F)$. The main theorem establishes that this…

We prove that there is a bound on the dimension of the first cohomology group of a finite group with coefficients in an absolutely irreducible in characteristic p in terms of the sectional p-rank of the group.

If $L(s,\pi)$ and $L(s,\rho)$ are the Dirichlet series attached to cuspidal automorphic representations $\pi$ and $\rho$ of ${\rm GL}_n({\mathbb A}_{\mathbb Q})$ and ${\rm GL}_{n-2}({\mathbb A}_{\mathbb Q})$ respectively, we show that…