We introduce a model for random groups in varieties of n-periodic groups as n-periodic quotients of triangular random groups. We show that for an explicit dcrit ∈ (1/3, 1/2), for densities d ∈ (1/3, dcrit) and for n large enough, the model produces infinite n-periodic groups. As an application, we obtain, for every fixed large enough n, for every p ∈ (1, ∞) an infinite n-periodic group with fixed points for all isometric actions on Lp-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindrically hyperbolic.
Bibliographical noteFunding Information:
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge for support and hospitality during the programme “Non-positive curvature, group actions and cohomology” where work on this paper was undertaken, supported by EPSRC grant EP/K032208/1. The research of the second author was also supported in part by EPSRC grant EP/P010245/1.
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