Laurent phenomenon algebras arising from surfaces II: Laminated surfaces

It was shown by Fock and Goncharov (Dual Teichmuller and lamination spaces. Handbook of Teichmuller Theory, 2007), and Fomin et al. (Acta Math 201(1):83-146, 2008) that some cluster algebras arise from triangulated orientable surfaces. Subsequently, Dupont and Palesi (J Algebraic Combinatorics 42(2):429-472, 2015) generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. In Wilson (Int Math Res Notices 341, 2017) we linked this framework to Lam and Pylyavskyy's Laurent phenomenon algebras (J Math 4(1):121-162, 2016), showing that unpunctured surfaces admit an LP structure. In this paper we extend quasi-cluster algebras to include punctured surfaces. Moreover, by adding laminations to the surface we demonstrate that all punctured and unpunctured surfaces admit LP structures. In short, we link two constructions which arose as seemingly unrelated generalisations of cluster algebras-one of the generalisations (quasi-cluster algebras) being based on triangulated surfaces, and the other (Laurent phenomenon algebras) based on the Laurent phenomenon. We thus provide a rich class of geometric examples in which to help study Laurent phenomenon algebras.