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Simplicial cohomology of augmentation ideals in $$\ell^1(G)$$. (English) Zbl 1264.43003
Introduction: 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 cohomology of $$G$$ into simplicial cohomology of $$\ell^1(G)$$ is an isomorphism.
The bounded cohomology of a (discrete) group $$G$$ is known to embed as a summand in the simplicial cohomology of the convolution algebra $$\ell^1(G)$$. Consequently, knowing that the bounded cohomology of $$G$$ is non-zero, or non-Hausdorff, immediately implies that the simplicial cohomology of $$\ell^1(G)$$ is non-zero or non-Hausdorff, respectively. In this paper we observe that for a wide class of discrete groups, including all torsion-free hyperbolic groups, this summand is the only non-zero contribution to simplicial cohomology; more precisely, the aforementioned inclusion of bounded cohomology into simplicial cohomology is an isomorphism. The precise statement is given as Theorem 3.5. By standard homological arguments (see Lemma 3.1) we may recast our result as saying that the augmentation ideals for these groups are simplicially trivial, in the sense that the ‘naive’ Hochschild cohomology groups $$\mathcal H^*(I_0(G),I_0(G)')$$ vanish: see Corollary 5.1). Thus, our work is a partial generalization of results of N. Grønbæk and A. T. Lau [Math. Proc. Camb. Philos. Soc. 126, No. 1, 139–148 (1999; Zbl 0918.43005)] on weak amenability of such ideals.
Our work is also motivated by A. Pourabbas and M. C. White [The second bounded group cohomology of group algebras, preprint], in which a version of our decomposition theorem is given for second-degree cohomology; there, the conclusion is stronger because the second bounded cohomology of any discrete group is known to be a Banach space (no such general result is true for degrees 3 and above).
##### MSC:
 43A20 $$L^1$$-algebras on groups, semigroups, etc. 22D15 Group algebras of locally compact groups 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46H05 General theory of topological algebras 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 20F67 Hyperbolic groups and nonpositively curved groups
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