Assembly maps for topological cyclic homology of group algebras

We use assembly maps to study , the topological cyclic homology at a prime of the group algebra of a discrete group with coefficients in a connective ring spectrum . For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infi...

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Bibliographic Details
Published in:Journal für die reine und angewandte Mathematik 2019-10, Vol.2019 (755), p.247-277
Main Authors: Lück, Wolfgang, Reich, Holger, Rognes, John, Varisco, Marco
Format: Article
Language:English
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Summary:We use assembly maps to study , the topological cyclic homology at a prime of the group algebra of a discrete group with coefficients in a connective ring spectrum . 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.
ISSN:0075-4102
1435-5345
DOI:10.1515/crelle-2017-0023