Groups with Finitely Many Shortlex Cones
arXiv:2607.14036
summary
The paper proves that any finitely generated group with the falsification‑by‑fellow‑traveller property also satisfies its shortlex version, yielding finitely many shortlex cone types and a regular language of shortlex representatives, and shows that the converse implications fail.
Abstract
We show that a finitely generated group 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 is a regular language. Following the example of M. Elder, we prove that the converse of the previous statements do not hold.
Topics & keywords
#finitely generated groups#falsification by fellow traveller property#shortlex cone types#regular languages#group theoryfalsification by fellow travellershortlex falsificationcone typesregular languageshortlex representatives