Two examples of vanishing and squeezing in \(K_1\). (English) Zbl 1454.19001
Summary: 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 H. Bass et al. [Publ. Math., Inst. Hautes Étud. Sci. 22, 545–564 (1964; Zbl 0248.18026)].
MSC:
| 19B28 | \(K_1\) of group rings and orders |
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
Citations:
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