Weighted growth functions of automatic groups
Author and article information
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.
Cite this as
2017. Weighted growth functions of automatic groups. PeerJ Preprints 5:e3256v1 https://doi.org/10.7287/peerj.preprints.3256v1Author comment
This is a submission to PeerJ Computer Science for review.
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Competing Interests
The author declare that they have no competing interests.
Author Contributions
Mikael Vejdemo-Johansson wrote the paper, prepared figures and/or tables, performed the computation work, reviewed drafts of the paper.
Data Deposition
The following information was supplied regarding data availability:
FigShare
DOI 10.6084/m9.figshare.5375431
Funding
The author received no funding for this work.
