Weighted growth functions of automatic groups
- Subject Areas
- Theory and Formal Methods
- automatic group, group theory, automata, formal grammar, growth function, cayley graph, grobner bases, finitely presented groups, symbolic computation
- © 2017 Vejdemo-Johansson
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Preprints) and either DOI or URL of the article must be cited.
- Cite this article
- 2017. Weighted growth functions of automatic groups. PeerJ Preprints 5:e3256v1 https://doi.org/10.7287/peerj.preprints.3256v1
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.
This is a submission to PeerJ Computer Science for review.