Algebraic K-theory over the infinite dihedral group: a controlled topology approach

TL;DR

This paper applies controlled topology to analyze algebraic K-theory over the infinite dihedral group, leading to simplifications in the Farrell-Jones conjecture and explicit computations of Nil groups.

Contribution

It introduces a controlled topology approach to simplify the Farrell-Jones conjecture and compute Waldhausen Nil groups for groups mapping onto the infinite dihedral group.

Findings

Reduction of the K-theoretic Farrell-Jones conjecture to specific virtually cyclic groups.

Explicit computation of Waldhausen Nil groups in terms of Farrell-Bass Nil groups.

Application of controlled topology to algebraic K-theory problems.

Abstract

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group which maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell-Bass Nil groups of the index two subgroup which maps surjectively to the infinite cyclic group.