Experiments on growth series of braid groups

TL;DR

This paper develops an algorithmic approach to study the growth series of braid groups with different generators, providing experimental data and conjectures for specific cases.

Contribution

It introduces a new framework for analyzing growth series of braid groups and offers conjectured rational formulas based on experimental results.

Findings

Experimental evidence for rational growth series in braid groups.

Conjectured formulas for spherical growth series with Birman-Ko-Lee generators.

Analysis limited to three and four strand braid groups.

Abstract

We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman-Ko-Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman-Ko-Lee's generators.