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Infinite rank surface cluster algebras.

Canakci, I. and Felikson, A. (2019) 'Infinite rank surface cluster algebras.', Advances in mathematics., 352 . pp. 862-942.


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.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2019 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:21 May 2019
Date deposited:03 June 2019
Date of first online publication:26 June 2019
Date first made open access:26 June 2020

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