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Random group actions on \(\textrm{CAT}(0)\) square complexes

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Abstract

Generalizing ideas in Jahncke (Zufallsgruppen und die eigenschaft FR. Master’s thesis, 2012), we introduce the notion of progression in \(\textrm{CAT}(0)\) square complexes. Using progression, we are able to build on the proof strategy of Dahmani et al. (Math Ann 349(3): 657–673, 2011) to show any action of a random group with seven or more generators on a \(\textrm{CAT}(0)\) square complex has a global fixed point.

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Notes

  1. Note that \(\frac{1}{5}\) is \(\frac{4}{5}\cdot \frac{1}{4}\), \(\frac{5}{24}\) is \(\frac{5}{6}\cdot \frac{1}{4}\), and \(\frac{3}{14}\) is \(\frac{6}{7}\cdot \frac{1}{4}\).

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Acknowledgements

I would like to express my gratitude to Piotr Przytycki, my advisor, without whom none of this work would be possible. Thank you for posing this problem to me. Thank you for our regular meetings. And thank you for encouraging me when the work was especially difficult. Thank you Adrien Abgrall for revising early drafts of this paper, and thanks to the anonymous referee for their careful reading and comments.

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  1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Zachary Munro

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  1. Zachary Munro
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Munro, Z. Random group actions on \(\textrm{CAT}(0)\) square complexes. Math. Z. 307, 1 (2024). https://doi.org/10.1007/s00209-024-03460-5

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