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Full Description
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors construct a geometric realization in terms of suitable decorated Teichmuller 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.
The authors' 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 the authors' previous paper, removing unnecessary assumptions on the surface.
Contents
Introduction
Non-normalized cluster algebras
Rescaling and normalization
Cluster algebras of geometric type and their positive realizations
Bordered surfaces, arc complexes, and tagged arcs
Structural results
Lambda lengths on bordered surfaces with punctures
Lambda lengths of tagged arcs
Opened surfaces
Lambda lengths on opened surfaces
Non-normalized exchange patterns from surfaces
Laminations and shear coordinates
Shear coordinates with respect to tagged triangulations
Tropical lambda lengths
Laminated Teichmuller spaces
Topological realizations of some coordinate rings
Principal and universal coefficients
Appendix A. Tropical degeneration and relative lambda lengths
Appendix B. Versions of Teichmuller spaces and coordinates
Bibliography
