Random groups arising as graph products

TL;DR

This paper investigates the hyperbolic properties of random groups formed as graph products from Erdős–Rényi random graphs, establishing threshold functions for hyperbolicity and analyzing automorphism groups of associated right-angled Artin groups.

Contribution

It provides new threshold functions for hyperbolicity in graph product groups and examines automorphism groups of random right-angled Artin groups.

Findings

Threshold functions for hyperbolicity in graph products

Finite outer automorphism groups for certain random right-angled Artin groups

High probability of specific automorphism group properties as n increases

Abstract

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdos - Renyi model of a random graph and find precise threshold functions for the hyperbolicity (or relative hyperbolicity). We aslo study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as $n\to \infty$, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter $p$ is constant and satisfies $0.2929 <p<1$.