New York Journal of Mathematics
Volume 26 (2020), 607-635


E. Ellis, E. Rodríguez Cirone, G. Tartaglia, and S. Vega

Two examples of vanishing and squeezing in K1

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Published: June 6, 2020.
Keywords: Assembly maps, controlled topology, Bass-Heller-Swan theorem.
Subject: 19B28,18F25.

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.


All authors were partially supported by grant ANII FCE-3-2018-1-148588. The first author is partially supported by ANII, CSIC and PEDECIBA. G. Tartaglia and S. Vega were supported by CONICET . The last three authors were partially supported by grants UBACYT 20020170100256BA and PICT 2017--1935.

Author information

E. Ellis:
IMERL, Facultad de Ingeniería
Universidad de la República, Montevideo, Uruguay


E. Rodríguez Cirone:
Dep. Matemática - FCEyN - UBA
Buenos Aires, Argentina


G. Tartaglia:
Dep. Matemática-CMaLP, FCE-UNLP
La Plata, Argentina


S. Vega:
Dep. Matemática - FCEyN - UBA, IMAS - CONICET
Buenos Aires, Argentina