Random triangular Burnside groups

Abstract

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.