Random Artin groups

TL;DR

This paper introduces a new probabilistic model for random Artin groups, analyzing how their properties vary with the growth rate of defining coefficients relative to group rank, and shows that many conjectures hold asymptotically.

Contribution

It presents a novel model controlling the growth of coefficients in Artin groups and demonstrates that most conjectures are satisfied under this model for various growth speeds.

Findings

Most conjectures about Artin groups hold asymptotically almost surely.

Probability calculations for belonging to specific Artin group families.

Different growth speeds lead to distinct asymptotic properties.

Abstract

We introduce a new model of random Artin groups. The two variables we consider are the rank of the Artin groups and the set of permitted coefficients of their defining graphs. The heart of our model is to control the speed at which we make that set of permitted coefficients grow relatively to the growth of the rank of the groups, as it turns out different speeds yield very different results. We describe these speeds by means of (often polynomial) functions. In this model, we show that for a large range of such functions, a random Artin group satisfies most conjectures about Artin groups asymptotically almost surely. Our work also serves as a study of how restrictive the commonly studied families of Artin groups are, as we compute explicitly the probability that a random Artin group belongs to various families of Artin groups, such as the classes of $2$-dimensional Artin groups, $FC$-type Artin groups, large-type Artin groups, and others.