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arXiv · 1907.06135

Two examples of vanishing and squeezing in $K_1$

Eugenia Ellis·Emanuel Rodríguez Cirone·Gisela Tartaglia·Santiago Vega
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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.

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Eugenia Ellis, Emanuel Rodríguez Cirone, Gisela Tartaglia, Santiago Vega. 2019-08-01. Two examples of vanishing and squeezing in $K_1$. https://arxiv.org/abs/1907.06135

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