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

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 that admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group that maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell–Bass Nil groups of the index 2 subgroup that maps surjectively to the infinite cyclic group.