Assembly maps for topological cyclic homology of group algebras

TL;DR

This paper investigates the assembly maps in topological cyclic homology of group algebras, proving isomorphism for finite groups and analyzing injectivity and surjectivity for infinite groups, with applications to hyperbolic and abelian groups.

Contribution

It establishes new results on the behavior of assembly maps in topological cyclic homology for various classes of groups, including isomorphism and injectivity properties.

Findings

Assembly map is an isomorphism for finite groups.

Pro-isomorphism and injectivity results for infinite groups.

Counterexamples to injectivity and surjectivity in general.

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.