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  • Finite presentability of twisted Brin–Thompson groups

  • Part of
  • Matt C. B. Zaremsky
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
    Published online by Cambridge University Press:
    18 December 2024, pp. 1-22
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  • Given a group G acting faithfully on a set S, we characterize precisely when the twisted Brin–Thompson group SVG is finitely presented. The answer is that SVG is finitely presented if and only if we have the following: G is finitely presented, the action of G on S has finitely many orbits of two-element subsets of S, and the stabilizer in G of any element of S is finitely generated. Since twisted Brin–Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone–Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest.