Cone types and geodesic languages for lamplighter groups and Thompson's group F

Cleary, Sean, Elder, Murray and Taback, Jennifer (2006) Cone types and geodesic languages for lamplighter groups and Thompson's group F. Journal of Algebra, 303 2: 476-500. doi:10.1016/j.jalgebra.2005.11.016


Author Cleary, Sean
Elder, Murray
Taback, Jennifer
Title Cone types and geodesic languages for lamplighter groups and Thompson's group F
Formatted title
Cone types and geodesic languages for lamplighter groups and Thompson's group F
Journal name Journal of Algebra   Check publisher's open access policy
ISSN 0021-8693
1090-266X
Publication date 2006-09-15
Sub-type Article (original research)
DOI 10.1016/j.jalgebra.2005.11.016
Volume 303
Issue 2
Start page 476
End page 500
Total pages 25
Place of publication Maryland Heights, MO, United States
Publisher Academic Press
Language eng
Subject 010105 Group Theory and Generalisations
Formatted abstract
We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and that there is no regular language of geodesics. We show that the existence of families of “seesaw” elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.
Keyword Regular language
Rational growth
Cone type
Context-free grammar
Counter automata
Lamplighter groups
Thompson's group F
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: Excellence in Research Australia (ERA) - Collection
School of Mathematics and Physics
 
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