Random triangular Burnside groups

TL;DR

This paper introduces a new model for random groups in varieties of n-periodic groups, demonstrating the existence of infinite n-periodic groups with fixed points in L^p-spaces and establishing their acylindrical hyperbolicity.

Contribution

It presents a novel model for random n-periodic groups via triangular quotients and proves their uniform acylindrical hyperbolicity, extending understanding of their geometric properties.

Findings

Existence of infinite n-periodic groups for densities below a critical threshold.

Construction of infinite n-periodic groups with fixed points in L^p-spaces.

Proof that certain random triangular groups are uniformly acylindrically hyperbolic.

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 $d_{\mathrm{crit}}\in(1/3,1/2)$, for densities $d\in(1/3,d_{\mathrm{crit}})$ and for $n$ large enough, the model produces \emph{infinite} $n$-periodic groups. As an application, we obtain, for every fixed large enough $n$, for every $p\in (1,\infty)$ 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.