Infinite rank surface cluster algebras

Canakci, I.; Felikson, A.

Abstract

We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation sequences. We show transitivity of infinite mutation sequences on triangulations of an infinite surface and examine different types of mutation sequences. Moreover, we use a hyperbolic structure on an infinite surface to extend the notion of surface cluster algebras to infinite rank by giving cluster variables as lambda lengths of arcs. Furthermore, we study the structural properties of infinite rank surface cluster algebras in combinatorial terms, namely we extend “snake graph combinatorics” to give an expansion formula for cluster variables. We also show skein relations for infinite rank surface cluster algebras.

Citation

Canakci, I., & Felikson, A. (2019). Infinite rank surface cluster algebras. Advances in Mathematics, 352, 862-942. https://doi.org/10.1016/j.aim.2019.06.008

• Journal Article Type: Article
• Acceptance Date: May 21, 2019
• Online Publication Date: Jun 26, 2019
• Publication Date: Aug 20, 2019
• Deposit Date: Jun 3, 2019
• Publicly Available Date: Jun 26, 2020
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 352
• Pages: 862-942
• DOI: https://doi.org/10.1016/j.aim.2019.06.008
• Public URL: https://durham-repository.worktribe.com/output/1295330
• Related Public URLs: https://arxiv.org/abs/1704.01826