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Agrarian and $L^2$-invariants

Volume 255 / 2021

Fabian Henneke, Dawid Kielak Fundamenta Mathematicae 255 (2021), 255-287 MSC: Primary 20J05; Secondary 12E15, 16S35, 20E06, 57Q10. DOI: 10.4064/fm808-4-2021 Published online: 9 September 2021

Abstract

We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope for finite free $G$-CW-complexes together with a fixed choice of a ring homomorphism from the group ring $\mathbb {Z} G$ to a skew field. For the particular choice of the Linnell skew field $\mathcal {D}(G)$, this approach recovers most of the information encoded in the corresponding $L^2$-invariants.

As an application, we prove that for agrarian groups of deficiency $1$, the agrarian polytope admits a marking of its vertices which controls the Bieri–Neumann–Strebel invariant of the group, improving a result of the second author and partially answering a question of Friedl–Tillmann.

We also use the technology developed here to prove the Friedl–Tillmann conjecture on polytopes for two-generator one-relator groups; the proof forms the content of another article.

Authors

  • Fabian HennekeMax-Planck-Institut für Mathematik
    Vivatsgasse 7
    53111 Bonn, Germany
    e-mail
  • Dawid KielakFakultät für Mathematik
    Universität Bielefeld
    Postfach 100131
    33501 Bielefeld, Germany
    and
    Mathematical Institute
    University of Oxford
    Andrew Wiles Building
    Radcliffe Observatory Quarter
    Woodstock Road
    Oxford OX2 6GG, United Kingdom
    e-mail

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