On the Hochschild homology of \(\ell^1\)-rapid decay group algebras (Q2035063)

Let \(G\) be a finitely generated discrete group. Let \(A\) be the \(\ell^1\)-rapid decay algebra of \(G\), which is defined using a length function on \(G\). Its elements are functions on \(G\) that remain absolutely summable when multiplied with polynomials in the length function, and its multiplication is convolution. The Hochschild and periodic cyclic homology of \(A\) is of interest in connection with the Bost assembly map, a variant of the Baum-Connes and Farrell-Jones assembly maps. Let \(\langle G\rangle\) be the set of conjugacy classes of \(G\). The Hochschild homology of \(A\) is naturally a module over the algebra \(\ell^\infty(\langle G\rangle)\) of bounded functions on \(\langle G\rangle\) with the pointwise product. This allows to define the restricted Hochschild homology \(\mathrm{HH}(A)_x\) of \(A\) at a conjugacy class \(x\in \langle G\rangle\) and a canonical map \(\mathrm{HH}(A) \to \prod_{x\in\langle G\rangle} \mathrm{HH}(A)_x\). This map is shown to be injective in this article for many semi-hyperbolic groups \(G\). This class contains hyperbolic groups and their central extensions, Artin groups of extra-large type, right-angled Artin groups, or groups that are hyperbolic relative to a finite collection of groups as listed above. In addition, the article identifies \(\mathrm{HH}(A)_x\) with the rapid-decay group homology of the centraliser of \(x\) under certain assumptions on the group and the conjugacy class. The proofs are based on explicit chain maps and homotopies on the Hochschild complex of the group ring \({\mathbb C}G\), which were used used to compute \(\mathrm{HH}({\mathbb C}G)\), and norm estimates for these maps on the rapid-decay algebra.