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 d_crit ∈ (1/3, 1/2), for densities d ∈ (1/3, d_crit) 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 L^p-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindrically hyperbolic.

中文翻译：

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随机三角形 Burnside 群

我们引入了一个模型，用于各种n周期群的随机群作为三角形随机群的n周期商。我们证明了一个明确d_暴击∈（1/3，1/2），对于密度ð ∈（1/3，d_暴击）和ñ足够大，该模型产生无限ñ为周期组。作为一个应用，我们得到，对于每个足够大的固定n，对于每个p ∈ (1, ∞) 无限n周期群，对于L ^p上的所有等距动作具有固定点-空格。我们的主要贡献是证明某些随机三角形群是一致的圆柱双曲线。