Random groups at density 𝑑<3/14 act non-trivially on a CAT(0) cube complex

Montee, MurphyKate

doi:10.1090/tran/8778

Random groups at density $d<3/14$ act non-trivially on a CAT(0) cube complex
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Trans. Amer. Math. Soc. 376 (2023), 1653-1682

Abstract:

For random groups in the Gromov density model at $d<3/14$, we construct walls in the Cayley complex $X$ which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities $d<1/5$ and $d<5/24$, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density $d<1/4$.

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