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Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

About this Title

Sergey Fomin and Dylan Thurston

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 255, Number 1223
ISBNs: 978-1-4704-2967-6 (print); 978-1-4704-4823-3 (online)
Published electronically: August 14, 2018
Keywords:Cluster algebra, lambda length, decorated Teichmüller space, opened surface, tagged triangulation, shear coordinates, integral lamination, Ptolemy relations

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Non-normalized cluster algebras
  • Chapter 3. Rescaling and normalization
  • Chapter 4. Cluster algebras of geometric type and their positive realizations
  • Chapter 5. Bordered surfaces, arc complexes, and tagged arcs
  • Chapter 6. Structural results
  • Chapter 7. Lambda lengths on bordered surfaces with punctures
  • Chapter 8. Lambda lengths of tagged arcs
  • Chapter 9. Opened surfaces
  • Chapter 10. Lambda lengths on opened surfaces
  • Chapter 11. Non-normalized exchange patterns from surfaces
  • Chapter 12. Laminations and shear coordinates
  • Chapter 13. Shear coordinates with respect to tagged triangulations
  • Chapter 14. Tropical lambda lengths
  • Chapter 15. Laminated Teichmüller spaces
  • Chapter 16. Topological realizations of some coordinate rings
  • Chapter 17. Principal and universal coefficients
  • Appendix A. Tropical degeneration and relative lambda lengths
  • Appendix B. Versions of Teichmüller spaces and coordinates


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 Teichmüller 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.

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