Groups with Finitely Many Shortlex Cones

Lucía Asencio-Martín; Paloma López-Larios

Groups with Finitely Many Shortlex Cones

Group Theory 2026-07-15 v1

Authors: Lucía Asencio-Martín , Paloma López-Larios

Abstract

We show that a finitely generated group $G=\langle\Sigma\rangle$ satisfying the falsification by fellow traveller property also satisfies the shortlex falsification by fellow traveller property. This implies that there are finitely many shortlex cone types and, therefore, that the corresponding language of shortlex representatives $\text{SL}_G\subset (\Sigma\cup\Sigma^{-1})^*$ is a regular language. Following the example of M. Elder, we prove that the converse of the previous statements do not hold.

Cite

@article{arxiv.2607.14036,
  title  = {Groups with Finitely Many Shortlex Cones},
  author = {Lucía Asencio-Martín and Paloma López-Larios},
  journal= {arXiv preprint arXiv:2607.14036},
  year   = {2026}
}