Infinite rank surface cluster algebras

İlke Çanakçı*, Anna Felikson

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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.

Original languageEnglish
Pages (from-to)862-942
Number of pages81
JournalAdvances in Mathematics
Volume352
DOIs
Publication statusPublished - 20 Aug 2019
Externally publishedYes

Keywords

  • Decorated Teichmüller space
  • Infinite sequence of mutations
  • Infinite triangulation
  • Lambda length
  • Skein relation
  • Surface cluster algebra

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