Cluster algebras and triangulated surfaces. Part II: Lambda lengths

Understand

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.