On the K- and L-theory of hyperbolic and virtually finitely generated abelian groups

We investigate the algebraic K- and L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.

💡 Research Summary

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The paper investigates the algebraic K‑theory and L‑theory of group rings (RG) when the group (G) is either a (possibly non‑elementary) hyperbolic group or a virtually finitely generated abelian group, and (R) is an arbitrary associative ring with unit. The central goal is to establish the Farrell–Jones Conjecture (FJC) for these classes of groups and to derive explicit decompositions of the resulting K‑ and L‑groups in terms of the K‑ and L‑theory of the coefficient ring (R) together with Nil‑ and UNil‑terms that arise from virtually cyclic subgroups.

Main results.

1. Hyperbolic groups. Using the controlled algebra techniques pioneered by Bartels, Lück, and Reich, together with the flow‑space construction for hyperbolic groups, the authors prove that the assembly maps
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