Cluster algebras and triangulated surfaces. Part II: Lambda lengths

TL;DR

This paper develops a geometric model for cluster algebras associated with bordered surfaces using decorated Teichmueller space and lambda lengths, providing new insights and proofs for their structural properties.

Contribution

It introduces a geometric realization of cluster algebras via decorated Teichmueller space, incorporating arbitrary coefficients and providing an intrinsic interpretation of cluster variables.

Findings

Constructed a geometric model using decorated Teichmueller space.

Interpreted cluster variables as lambda lengths of arcs.

Provided alternative proofs for structural results.

Abstract

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.