Topology of Random Right Angled Artin Groups

TL;DR

This paper investigates the topological invariants of random right angled Artin groups, focusing on Betti numbers, cohomological dimension, and topological complexity, revealing probabilistic bounds on their values.

Contribution

It provides new probabilistic results on the topological complexity of random right angled Artin groups using combinatorial graph properties.

Findings

Topological complexity of random right angled Artin groups tends to at most three values.

The study links topological invariants to properties of random graphs.

Existence of bi-cliques in random graphs influences topological invariants.

Abstract

In this paper we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting complexity of motion planning algorithms in robotics. We show that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values. We use a result of Cohen and Pruidze which expresses the topological complexity of right angled Artin groups in combinatorial terms. Our proof deals with the existence of bi-cliques in random graphs.