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Geometric structures related to the braided Thompson groups

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Abstract

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type \({{\,\mathrm{F}\,}}_\infty \). The proof utilized certain contractible cube complexes, which in this paper we prove are \({{\,\mathrm{CAT}\,}}(0)\). We then use this fact to compute the geometric invariants \(\Sigma ^m(F_{{\text {br}}})\) of the pure braided Thompson group \(F_{{\text {br}}}\). Only the first invariant \(\Sigma ^1(F_{{\text {br}}})\) was previously known. A consequence of our computation is that as soon as a subgroup of \(F_{{\text {br}}}\) containing the commutator subgroup \([F_{{\text {br}}},F_{{\text {br}}}]\) is finitely presented, it is automatically of type \({{\,\mathrm{F}\,}}_\infty \).

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Notes

  1. The notation BG often stands for a classifying space of a group G, so even though there is not actually any risk of confusion we will stick to writing \(V_{{\text {br}}}\) and \(F_{{\text {br}}}\) instead of BV and BF.

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Acknowledgements

Thanks are due to Javier Aramayona and Rodrigo de Pool for catching a mistake in a previous version of this paper.

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  1. Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY, 12222, USA

    Matthew C. B. Zaremsky

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  1. Matthew C. B. Zaremsky
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Correspondence to Matthew C. B. Zaremsky.

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Zaremsky, M.C.B. Geometric structures related to the braided Thompson groups. Math. Z. 300, 2591–2610 (2022). https://doi.org/10.1007/s00209-021-02866-9

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