Algebraic K-theory over the infinite dihedral group: an algebraic approach

TL;DR

This paper establishes a connection between algebraic K-theory invariants of certain group constructions and extends the Farrell-Jones Conjecture to a broader class of subgroups, using algebraic methods.

Contribution

It provides an algebraic proof relating Waldhausen and Farrell-Bass nilpotent class groups for infinite dihedral groups and refines the Farrell-Jones Conjecture to finite-by-cyclic subgroups.

Findings

Isomorphism between Waldhausen and Farrell-Bass nilpotent class groups

Extension of the Farrell-Jones Conjecture to finite-by-cyclic subgroups

Algebraic approach to K-theory over the infinite dihedral group

Abstract

We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.