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Register a new userTrivial cup products in bounded cohomology of the free group via aligned chains
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We prove that the cup product of $Delta$-decomposable quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial. As a corollary we obtain that this is also the case for Brooks quasimorphisms (in particular on selfoverlapping words) and Rolli quasimorphisms.
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This paper is devoted to the computation of the space $H_b^2(Gamma,H;mathbb{R})$, where $Gamma$ is a free group of finite rank $ngeq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $Gamma$ if and only if $H_b^2(Gamma,H;mathbb{R})$ is not trivial, and furthermore, if and only if there is an isometric embedding $oplus_infty^nmathcal{D}(mathbb{Z})hookrightarrow H_b^2(Gamma,H;mathbb{R})$, where $mathcal{D}(mathbb{Z})$ is the space of bounded alternating functions on $mathbb{Z}$ equipped with the defect norm.
We construct cup products of two different kinds for Hopf-cyclic cohomology. When the Hopf algebra reduces to the ground field our first cup product reduces to Connes cup product in ordinary cyclic cohomology. The second cup product generalizes Connes-Moscovicis characteristic map for actions of Hopf algebras on algebras.
This is the first one in a series of two papers on the continuation of our study in cup products in Hopf cyclic cohomology. In this note we construct cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and coalgebras. In the next paper we consider producing Hopf cyclic cocycle from equivariant Hopf cyclic cocycles. Our approach in both situations is based on (co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilber theorem which is different from the old definition of cup products defined via traces and cotraces on DG algebras and coalgebras.
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${rm Out}(F_n)$, dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.
We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible (resp. indecomposable) $mathbb{K}_{par}G$-module with the ordinary homology and cohomology groups of $G$ with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension $cd_{ mathbb{K}}(G)$ (over a field $mathbb{K}$) with the partial cohomological dimension $cd_{ mathbb{K}}^{par}(G)$ (over $mathbb{K}$) and show that $cd_{ mathbb{K}}^{par}(G) geq cd_{ mathbb{K}}(G)$ and that there is equality for $G = mathbb{Z}$.
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