# Two examples of vanishing and squeezing in $K_1$

## TL;DR

This paper explores controlled topology techniques in algebraic K-theory, demonstrating vanishing and squeezing results for specific groups, notably infinite cyclic and dihedral groups, with implications for the isomorphism conjecture.

## Contribution

It provides explicit squeezing methods for generators in K_1 for these groups, extending known results and illustrating the machinery in concrete examples.

## Key findings

- Vanishing theorem for K_1 in the given group contexts
- Explicit squeezing of generators in K_1
- Extension of Bass-Heller-Swan results to these cases

## 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 $K_1$. For the infinite cyclic group, when taking coefficients in a regular ring, we get a squeezing result for every element of $K_1$; this follows from the well-known result of Bass, Heller and Swan.