Weighted growth functions of automatic groups

TL;DR

This paper introduces a new method to compute weighted growth functions of automatic groups using context-free grammars, enabling analysis of how different generator weights affect group growth.

Contribution

It presents a novel approach to calculate weighted growth functions for automatic groups via algebraic systems derived from context-free grammars.

Findings

Recovered known growth functions for small braid groups

Calculated weighted growth functions for braid groups with generator length weights

Demonstrated the method's ability to incorporate diverse weightings into growth analysis

Abstract

The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign different or even distinct weights to each of the generators in an underlying presentation, such that this weighting is reflected in the growth function. We recover known growth functions for small braid groups, and calculate growth functions that weight each generator in an automatic presentation of the braid groups according to their lengths in braid generators.