Assembly maps for topological cyclic homology of group algebras

Abstract

We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is injective but in general not surjective.

Final version, to appear in Crelle's Journal. Corrections to Theorems 1.5(ii), 1.8(ii), and Question 4.9. Improvements to the exposition. 30 pages

References in corpus (3)

• Algebraic K-theory of group rings and the cyclotomic trace map
• Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods
• Orbit categories, classifying spaces, and generalized homotopy fixed points