Two examples of vanishing and squeezing in K₁

Abstract

Controlled topology is one of the main tools for proving the isomorphism conjecture concerning the algebraic K-theory of group rings. In this article we dive into this machinery in two examples: when the group is infinite cyclic and when it is the infinite dihedral group in both cases with the family of finite subgroups. We prove a vanishing theorem and show how to explicitly squeeze the generators of these groups in K₁. For the infinite cyclic group, when taking coefficients in a regular ring, we get a squeezing result for every element of K₁; this follows from the well-known result of Bass, Heller and Swan.